Scaling of quasibrittle fracture: asymptotic analysis

Fracture of quasibrittle materials such as concrete, rock, ice, tough ceramics and various fibrous or particulate composites, exhibits complex size effects. An asymptotic theory of scaling governing these size effects is presented, while its extension to fractal cracks is left to a companion paper [1] which follows. The energy release from the structure is assumed to depend on its size D, on the crack length, and on the material length c _f governing the fracture process zone size. Based on the condition of energy balance during fracture propagation and the condition of stability limit under load control, the large-size and small-size asymptotic expansions of the size effect on the nominal strength of structure containing large cracks or notches are derived. It is shown that the form of the approximate size effect law previously deduced [2] by other arguments can be obtained from these expansions by asymptotic matching. This law represents a smooth transition from the case of no size effect, corresponding to plasticity, to the power law size effect of linear elastic fracture mechanics. The analysis is further extended to deduce the asymptotic expansion of the size effect for crack initiation in the boundary layer from a smooth surface of structure. Finally, a universal size effect law which approximately describes both failures at large cracks (or notches) and failures at crack initiation from a smooth surface is derived by matching the aforementioned three asymptotic expansions. Walter P. Murphy Professor of This revised version was published online in July 2006 with corrections to the Cover Date.     

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.     

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).     

Scaling and fractality in fatigue resistance: Specimen‐size effects on Wöhler's curve and fatigue limit

The present contribution investigates size effects on Wöhler's curve in accordance with dimensional analysis and intermediate asymptotics theory. These approaches provide a generalised equation able to interpret the specimen‐size effects on Wöhler's curve. Subsequently, using a different approach based on lacunar fractality concepts, analogous scaling laws are found for the coordinates of the limit‐points of Wöhler's curve, so that a theoretical explanation is provided to the decrement in fatigue resistance by increasing the specimen size. Eventually, the proposed models are compared with experimental data available in the Literature, which seem to confirm the advantage of applying fractal geometry to the problem.     

Multi-range Fractals in Materials

A new model of multi-range fractals is proposed to explain the experimental results observed on thefractal dimensions of the fracture surfaces in materials.The relationship of multi-range fractals withmulti-scaling fractals has been also discussed.     

Generalizations and specializations of cohesive crack models

This paper presents an overview of the cohesive crack model, one of the basic models used so far to describe the fracture of concrete and other quasibrittle materials. Recent developments and needs for further research are discussed. The displayed evidence and the discussion are based on considering the cohesive crack model as a constitutive assumption rather than an ad hoc model for the behaviour ahead of a preexisting crack. Topics addressed are fracture of unnotched specimens, mixed mode fracture, diffuse cracking, anomalous stress-strain curves, size effect and asymptotic analysis, and strength of structural elements with notches.     

General size effect on strength of bimaterial quasibrittle structures

This paper presents a general size effect equation for the strength of hybrid structures, which are made of two dissimilar quasibrittle materials with a thin and weak bimaterial interface. Depending on the material mismatch and structure geometry, a singular stress field could occur at the bimaterial corner. For structures with strong stress singularities, an energetic size effect is derived based on the equivalent linear elastic fracture mechanics and asymptotic matching. For structures without stress singularities, a finite weakest link model is adopted to derive the size effect. A general scaling equation that bridges the limits of strong and zero stress singularities is formulated by combining the energetic scaling of fracture of the bimaterial corner and the finite weakest link model.     

On Fractals and Size Effects

Following an extensive and critical review of fractals and size effects models, this paper seeks to generalize Bažant’s size effect law to fractal cohesive cracks. This is achieved through a Newtonian approach in which the cohesive and far field stress intensity factors of fractal cracks (derived by Yavari) are set equal. It will be shown that the fractal size effect law is a generalization of the one of Bažant (derived in Euclidian space). In light of the derived equation, the multi-fractal model of Carpinteri and the size effect law of Bažant are revisited. Finally, the paper will conclude with some general considerations pertaining to the so-called “New Kind of Science” developed by Wolfram, and its applicability to fracture mechanics.     

Response to A. Carpinteri, B. Chiaia, P. Cornetti and S. Puzzi’s Comments on “Is the cause of size effect on structural strength fractal or energetic-statistical?”

Carpinteri et al.’s discussion is very welcome for it gives an opportunity to clarify long-running disagreements on the problem of size effect, important to several engineering fields. However, the discussion misinterprets many points of Ba ant and Yavari’s paper and attempts to raise new issues. This response presents recent experimental results contradicting applicability of Carpinteri’s “multifractal scaling law” (MFSL), and refutes the discussers’ arguments on their proposed concepts of “fractal mechanics”, on the statistical size effect, on the validity of mathematical derivation of MFSL and its asymptotic slope, and on various other aspects of scaling of quasibrittle failure.     

Multirange Fractal Analysis on the Negative Correlation between Fractal Dimension of Fractured Surface and Toughness of Materials

Applying the concept of multirange fractals, a new explanation to the Williford's multifractal curve on the relationship of fractal dimension with fracture properties in materials has been given. It 5hows the importance of factorizing out the effect of fractal structure from other physical causes and separating the appropriate range of scale from multirange fractals     

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.     

Crack propagation: Dynamic stochasticity and scaling

The results of theoretical and experimental investigations provide explanation of the mechanism of stochastic branching during crack propagation in a solid, relating this phenomenon to changes in the fundamental symmetry relationships in a nonlinear system representing the solid with defects. The character of the stochastic behavior of quasibrittle materials is determined by nonlinear dynamics of the ensemble of microscopic defects under the conditions of kinetic transitions. The transition from dispersed to macroscopic fracture is accompanied by the appearance of multiple fracture zones in the regime of explosion-like instability development over a discrete spectrum of spatial scales.     

Determination for the time-to-fracture of solids

A method to determine the time to fracture taking into account the physical mechanisms of microcracks and crack formation is developed on the basis of the fractal model of fracture. The fractal dimension of a crack at different stages of its growth is determined theoretically. The damage evolution law which allows for the kinetic and microstructural properties of a material is obtained on the basis of the kinetic theory of strength. Conditions at which the microcracks accumulation gives way to the propagation of a large crack are determined with the use of the percolation theory. It is shown that the fractal dimension of the initial part of a crack is much more than the fractal dimension of the rest of the crack.     

Simulation of microfracture process of brittle polycrystals: Microcracking and crack propagation

Microcracking and crack propagation behavior are simulated for 2-dimensional alumina polycrystals which have thermal anisotropy within a grain. Microcracks are generated by thermally induced residual stresses at the grain boundary. Stress redistribution due to microcracking and stress intensity factors at the microcrack tip are obtained numerically by the body force method. The location at which microfracture occurs is determined by a competition between microcracking and crack propagation under external stresses. The microfracture stress increases with the progress of fracture and decreases after the maximum indicating a fracture strength. In many cases, the extension of microcracks induces an unstable fracture. With both increasing grain size and decreasing grain boundary toughness, the number of microcracks prior to the unstable state increases and the stress concentration due to the microcracks plays a significant role in the stable crack extension, resulting in lower strengths than the fracture-mechanical predictions.     

无机膜的分形性

介绍了无机膜的分形性和物体的三种重要分形．在溶胶－凝胶（Ｓｏｌ－Ｇｅｌ）制无机膜的工艺中，胶粒通过生长聚集方式成膜，它受质量分形规律控制，可以通过测定质量分形维数，利用ＤＬＡ、ＤＬＣＡ、ＲＬＣＡ等数学模型定量描述Ｓｏｌ－Ｇｅｌ膜生长的形态构造及对孔结构的影响．同样，在注浆及浸涂工艺制微孔基质膜工艺中，粉体以桥接堆积，存在孔结构分形，可通过测定孔分形维数定量描述这种在三维空间下高度复杂的孔结构．     

Probability distribution of energetic-statistical size effect in quasibrittle fracture

The physical sources of randomness in quasibrittle fracture described by the cohesive crack model are discussed and theoretical arguments for the basic form of the probability distribution are presented. The probability distribution of the size effect on the nominal strength of structures made of heterogeneous quasibrittle materials is derived, under certain simplifying assumptions, from the nonlocal generalization of Weibull theory. Attention is limited to structures of positive geometry failing at the initiation of macroscopic crack growth from a zone of distributed cracking. It is shown that, for small structures, which do not dwarf the fracture process zone (FPZ), the mean size effect is deterministic, agreeing with the energetic size effect theory, which describes the size effect due to stress redistribution and the associated energy release caused by finite size of the FPZ formed before failure. Material randomness governs the statistical distribution of the nominal strength of structure and, for very large structure sizes, also the mean. The large-size and small-size asymptotic properties of size effect are determined, and the reasons for the existence of intermediate asymptotics are pointed out. Asymptotic matching is then used to obtain an approximate closed-form analytical expression for the probability distribution of failure load for any structure size. For large sizes, the probability distribution converges to the Weibull distribution for the weakest link model, and for small sizes, it converges to the Gaussian distribution justified by Daniels' fiber bundle model. Comparisons with experimental data on the size-dependence of the modulus of rupture of concrete and laminates are shown. Monte Carlo simulations with finite elements are the subject of ongoing studies by Pang at Northwestern University to be reported later.     

A discrete cohesive model for fractal cracks

The fractal crack model described here incorporates the essential features of the fractal view of fracture, the basic concepts of the LEFM model, the concepts contained within the Barenblatt-Dugdale cohesive crack model and the quantized (discrete or finite) fracture mechanics assumptions proposed by Pugno and Ruoff [Pugno N, Ruoff RS. Quantized fracture mechanics. Philos Mag 2004;84(27):2829-45] and extended by Wnuk and Yavari [Wnuk MP, Yavari A. Discrete fractal fracture mechanics. Engng Fract Mech 2008;75(5):1127-42]. The well-known entities such as the stress intensity factor and the Barenblatt cohesion modulus, which is a measure of material toughness, have been re-defined to accommodate the fractal view of fracture.For very small cracks or as the degree of fractality increases, the characteristic length constant, related to the size of the cohesive zone is shown to substantially increase compared to the conventional solutions obtained from the cohesive crack model. In order to understand fracture occurring in real materials, whether brittle or ductile, it seems necessary to account for the enhancement of fracture energy, and therefore of material toughness, due to fractal and discrete nature of crack growth. These two features of any real material appear to be inherent defense mechanisms provided by Nature.     

Influence of the crack morphology on the fatigue crack growth rate: A continuously-kinked crack model based on fractals

Threshold condition and rate of fatigue crack growth appear to be significantly affected by the degree of deflection of cracks. In the present paper, the reduction of the fatigue crack growth rate for a so-called ‘periodically-kinked crack’ as compared to that for a straight counterpart is quantified via the Paris–Erdogan law modified according to some simple theoretical arguments. It is shown that such a reduction increases as the value of the kinking angle increases. Then, a so-called ‘continuously-kinked crack’ (the kink length tends to zero) is considered and modelled as a self-similar invasive fractal curve. The sequence of kinking angles in the crack is such that the fatigue crack path is ‘on average’ straight. Using the Richardson’s expression for self-similar fractals, the fractal dimension of the crack is expressed as a function of the kinking angle. It is shown that the fatigue crack growth rate in the Paris range depends not only on the above fractal dimension and in turn on the kinking angle, but also, in an explicit fashion, on the crack length. Some experimental results related to concrete and showing a crack size effect on the fatigue crack growth rate are analysed.