An approach to size effect in fatigue of metals using fractal theories

ABSTRACT As was experimentally observed by several authors, the fatigue strength of metallic materials decreases with increasing the specimen size. Such a decrease can be remarkable for very large structures like, for example, big cargo ships (some hundred meters long) transporting oil or other goods. Size effect in fatigue is herein explained by considering the fractal nature of the reacting cross sections of structures, that is, the renormalized fatigue strength is represented by a force amplitude acting on a surface with a fractal dimension lower than 2, where such a dimensional decrement depends on a self‐similar weakening of the material ligament, owing to the presence of cracks, defects, voids and so forth (microscopic level). However, this decrement tends to progressively disappear with increasing the structure size (macroscopic level), i.e. the effect of the material microstructure on the macroscopic fatigue behaviour gradually vanishes for structures large enough with respect to a characteristic microstructural size, this phenomenon being defined as multifractality. A multifractal scaling law for fatigue limit of metals is proposed, and some experimental results are examined in order to show how to apply the theoretical approach presented.     

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.     

Using torsional bar testing to determine fracture toughness

A new method to determine fracture toughness K _IC of materials is introduced. A round-rod specimen having a V-grooved spiral line with a 45° pitch is tested under pure torsion. An equibiaxial tensile/compressive stress state is effectively created to simulate conventional test methods using a compact-type specimen with a thickness equivalent to the full length of the spiral line. K _IC values are estimated from the fracture load and crack length with the aid of a three-dimensional finite element analysis. K _IC of 7475-T7351 aluminium is estimated to be 51.3 MPa √m, which is higher than the vendor's value in the TL orientation by ∼0.8% and higher than 0.5T compact tension (CT) value by 6%; A302B steel yields 54.9 MPa √m being higher than CT test value by ∼2%. Good agreement between the K _IC values obtained by different methods indicates the proposed method is sound and reliable.     

The mechanics of self-similar and self-affine fractal cracks

In this paper we study the mechanical attributes of the fractal nature of fracture surfaces. The structure of stress and strain singularity at the tip of a fractal crack, which can be self-similar or self-affine, is studied. The three classical modes of fracture and the fourth mode of fracture are discussed for fractal cracks in two-dimensional and three- dimensional solid bodies. It is discovered that there are six modes of fracture in fractal fracture mechanics. The J-integral is shown to be path-dependent. It is explained that the proposed modified J-integrals in the literature that are argued to be path-independent are only locally path-independent and have no physical meaning. It is conjectured that a fractal J-integral should be the rate of potential energy release per unit of a fractal measure of crack growth. The powers of stress and strain singularities at the tip of a fractal crack in a strain-hardening material are calculated. It is shown that stresses and strains have weaker singularities at the tip of a fractal crack than they do at the tip of a smooth crack.     

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.     

Self-affinity of fracture surfaces and implications on a possible size effect on fracture energy

As demonstrated within the last 15 years by numerous experimental studies, tensile fracture surfaces exhibit a self-affine fractal geometry in many different materials and loading conditions. In the last few years, some authors proposed to explain an observed size effect on fracture energy by this fractality. However, because they did not consider a lower bound to this scale invariance (which necessarily exists, at least at the atomic scale), they had to introduce a new definition of fracture energy with unconventional physical dimensions. Moreover, they were unable to reproduce the observed asymptotic behavior of the apparent fracture energy at large specimen sizes. Here, we show that this is because they considered self-similar fracture surfaces (not observed in nature) instead of self-affine. It is demonstrated that the ignorance of the self-affine roughness of fracture surfaces when estimating the fracture energy from the work spent to crack a specimen necessarily leads, if the work of fracture is proportional to the fracture area created, to a size effect on this fracture energy. Because of the self-affine (instead of self-similar) character of fracture surfaces, this size effect follows an asymptotic behavior towards large scales. It is therefore rather limited and not likely detectable for relatively large sample sizes (10^–1 m). Consequently, significant and rapid increases of the apparent fracture energy are more likely to be explained mainly by other sources of size effect.     

复合型裂纹起裂角厚度效应的实验研究

完成了飞机结构铝合金LC4CS的2、4、8和14mm四种不同厚度试样在Ⅰ+Ⅱ复合加载条件下的复合型断裂实验,系统分析了厚度和复合载荷对裂纹起裂角的影响,揭示了常用复合型断裂准则的厚度适用范围,用三维断裂理论对结果进行了讨论.结果表明:复合型裂纹起裂角具有明显的厚度效应;最大周向应力准则能够准确预测薄试样和厚试样(厚度为2 mm 和14 mm)在各种复合加载条件下的起裂方向,但是不适用于中间厚度的试样,尤其是8 mm厚度的情况.最大三轴应力准则试图考虑裂纹尖端三维约束对裂纹起裂的影响,但是结果并不理想.最小应变能密度因子理论的预测结果与最大周向应力准则的预测结果非常接近,但同样不能预测8mm厚度试样的起裂方向.非常有必要建立一个普遍适用的三维复合型断裂准则.     

滑石粉填充PP材料中颗粒分布分形特征及其与冲击性能的关系

考察了滑石粉填充PP材料颗料分布的分形特性,提出了用分形维数D定量描述复合材料第二相颗粒在基体中的分散状况,通过对比D与冲击强度的关系,发现不同填充体系均存在一临界Dc值,在此分形维数下冲击强度发生转折。同时,通过测定材料冲击断面的粗糙度参量Rs,结果显示Rs同样也会发生与冲击强度相对应的转折变换。     

On the failure of cracks under mixed-mode loads

Fracture of plates containing a crack under mixed-mode, I and II, loading conditions is investigated. Fracture mechanisms are first examined from fracture surface morphology to correlate with the macroscopic fracture behavior. Two distinct features are observed and they are typical of shear and tensile types of failure. From this correlation, a fracture criterion based on the competition of the attainment of a tensile fracturing stress σ_{_C} and a shear fracturing stress τ_{_C} at a fixed distance around the crack tip is proposed. Material ductility is incorporated using τ_{_C}/σ_{_C} determined from classical material failure theories. The type of fracture is predicted by comparing τ_{_max}/σ_{_max} at r=r_{_C} for a given mixed mode loading to the material ductility_{τ_C/σ_C} , i.e. τ_{_max}/σ_{_max})<(τ_{_C} /σ_{_C}) for tensile type of fracture and (τ_{_max}/σ_{_max}) r (τ_{_C}/ σ_{_C}) for shear type of fracture. It is found that, for typical engineering structural metals with certain ductility, (1) crack propagation initiates according to the maximum hoop stress criterion when the the mode mixity is near mode I and according to the maximum shear stress criterion when the mode mixity is near mode II, and (2) the transition of the failure from tensile to shear type can be predicted by the proposed criterion. For brittle materials the maximum hoop (opening) stress always reaches the tensile fracturing stress before the maximum shear stress reaches the shear fracturing stress of the material at a crack tip. Therefore, specimens made of brittle materials tend to fail under the maximum hoop stress criterion, as demonstrated by Erdogan and Sih (1963) and others. This revised version was published online in August 2006 with corrections to the Cover Date.     

The role of T-stress in brittle fracture for linear elastic materials under mixed-mode loading

The purpose of this paper is to revisit the maximum tensile stress (MTS) criterion to predict brittle fracture for mixed mode conditions. Earlier experimental results for brittle fracture of polymethylmethacrylate (PMMA) using angled cracked plates are also re-examined. The role of the T -stress in brittle fracture for linear elastic materials is emphasized. The generalized MTS criterion is described in terms of mode I and II stress intensity factors, K _I and K _II and the T- stress (the stress parallel to the crack), and a fracture process zone, r _c . The generalized MTS criterion is then compared with the earlier experimental results for PMMA subjected to mixed mode conditions. It is shown that brittle fracture can be controlled by a combination of singular stresses (characterized by K ) or non-singular stress ( T -stress). The T -stress is also shown to have an influence on brittle fracture when the singular stress field is a result of mode II loading.     

Quantification of constraint effects in fracture mechanism transition for cracked structures under mixed mode loading

The objective of the study is to evaluate the effects of plastic constraint on transition between tensile-type and shear-type fracture. The T -stress is employed as the quantifying parameter for constraint and is incorporated into the existing theoretical criteria for modelling this transition. It is found that different constraint levels can dramatically alter the transition point. To verify this finding, two sets of mixed mode tests with different constraint levels are carried out. Alongside the theoretical and experimental study, finite element simulation is performed to verify and support these findings. Substantially improved agreement is observed with experimental data if the effect of plastic constraint on transition is included.     

Scaling phenomena due to fractal contact in concrete and rock fractures

Concrete-to-concrete friction contributes in many cases to the stability of a structure. At different scales, the slope stability of rock joints is deeply influenced by the surface morphology and shows a marked size-dependence. In this paper, the closure and sliding-dilatant behaviour of cracks in concrete and rocks is investigated by means of a coupled numerical/experimental approach. These natural interfaces show self-affine properties in the relevant scale range. Attention has been focused on the stress transfer mechanism across the interfaces, showing that the sets of contact points possess the self-similar character of lacunar fractal sets. Scaling laws come into play and the size-effects on the shear strength of rough interfaces, and on their closure deformability, can be explained. This revised version was published online in July 2006 with corrections to the Cover Date.     

A modified specimen for evaluating the mixed mode fracture toughness of adhesives

This article introduces a specimen geometry that allows the separation of fracture energy release rates G_I and G_II in adhesively joined beams made of disparate materials. The analysis is based upon a Green's functions formulation for shear deformable beams and circumvents the need to employ finite element computations. The current method results in a system of non-singular integral equations, that can be discretized and reduced to a system of algebraic equations which may be solved by common numerical techniques. The analysis accounts for the dimensions and properties of the adhesive and provides results for a wide range of G_I, G_II and their ratio. Those results agree with finite element computational values to within less than 4%.     

考虑骨料尺寸的混凝土岩石边界效应断裂模型

该文比较了边界效应模型（BEM）和尺寸效应模型（SEM）在研究材料断裂性能方面的不同。提出了由处于准脆性断裂状态的三点弯曲试件的峰值荷载P_max,同时确定材料参数--断裂韧度K_IC与拉伸强度f_t的理论与方法。由于实验室条件下混凝土试件高度W与骨料最大粒径d_max的比例W/d_max约为5~20,试件的非均质性明显,破坏为准脆性断裂控制。因此,区别于以连续介质力学为基础的应用于准脆性断裂研究的力学模型,该文研究将骨料最大粒径d_max引入相应的断裂模型解析表达式中,由参数组合β d_max来计算结构峰值状态对应的裂缝扩展量,通过离散参数β的不同取值,实现了对材料参数--断裂韧度与拉伸强度的准确预测。基于不同学者的相同尺寸W而不同初始裂缝长度a₀,以及相同初始缝高比a₀/W而不同尺寸W的几何相似的砂浆、混凝土及岩石类材料试件的试验成果（骨料最大粒径d_max从1.2 mm~40 mm变化）,验证了所提理论与方法的合理性。     

Invariant Points on Energy Contours Around a Crack Tip Under Mixed Mode Loading

The locus of points around a crack tip where the strain energy density is set equal to a critical value reveals interesting features. It is seen that, for certain cases, two points on this locus remain invariant with respect to the phase of the applied loads. The existence of these invariant points is examined for different configurations - a crack in a homogeneous isotropic medium, an interface crack and an inclined interface crack. This analysis is extended to the two components of strain energy density – volumetric (VSED) and distortional (DSED). This revised version was published online in August 2006 with corrections to the Cover Date.