Scaling of quasibrittle fracture: hypotheses of invasive and lacunar fractality, their critique and Weibull connection

Considerable progress has been achieved in fractal characterization of the properties of crack surfaces in quasibrittle materials such as concrete, rock, ice, ceramics and composites. Recently, fractality of cracks or microcracks was proposed as the explanation of the observed size effect on the nominal strength of structures. This explanation, though, has rested merely on intuitive analogy and geometric reasoning, and did not take into account the mechanics of crack propagation. In this paper, the energy-based asymptotic analysis of scaling presented in the preceding companion paper in this issue [1] is extended to the effect of fractality on scaling. First, attention is focused on the propagation of fractal crack curves (invasive fractals). The modifications of the scaling law caused by crack fractality are derived, both for quasibrittle failures after large stable crack growth and for failures at the initiation of a fractal crack in the boundary layer near the surface. Second, attention is focused on discrete fractal distribution of microcracks (lacunar fractals), which is shown to lead to an analogy with Weibull's statistical theory of size effect due to material strength randomness. The predictions ensuing from the fractal hypothesis, either invasive or lacunar, disagree with the experimentally confirmed asymptotic characteristics of the size effect in quasibrittle structures. It is also pointed out that considering the crack curve as a self-similar fractal conflicts with kinematics. This can be remedied by considering the crack to be an affine fractal. It is concluded that the fractal characteristics of either the fracture surface or the microcracking at the fracture front cannot have a significant influence on the law of scaling of failure loads, although they can affect the fracture characteristics. Walter P. Murphy, Professor| of This revised version was published online in July 2006 with corrections to the Cover Date.     

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).     

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.     

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 shear parameters on mixed–mode fracture of concrete

The influence of the mode II fracture parameters on the mixed mode fracture experimental tests of quasibrittle materials is studied. The study is based on experimental results and numerical analyses. For the numerical study, a procedure for mixed mode fracture of quasibrittle materials is presented. The numerical procedure is based on the cohesive crack approach, and extends it to mixed mode fracture. Four experimental sets of mixed mode fracture were modelled, one from Arrea and Ingraffea and another from a nonproportional loading by the authors, both with bending concrete beams. Two other sets of experimental fracture were modelled, based on double-edge notched testing; in these tests an important mode II is beforehand expected. The numerical results agree quite well with experimental records. The influence of the main parameters for mode II fracture on the mixed mode fracture is studied for the four experimental set of tests and compared with these results. In all them, large changes in the mode II fracture energy hardly modify the numerical results. The tangential and normal stresses along the crack path during the loading proccess are obtained, also with different values of the mode II fracture energy. For the studied experimental tests it is concluded that the crack is initiated under mixed mode but propagated under predominant mode I. This allows a development of mixed mode fracture models, mainly based on standard properties of the material measured by standard methods, avoiding the problems associated with the measurement of mode II fracture parameters, such as mode II fracture energy and cohesion.     

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.     

Nature of Microcracks in Ferritic Steels Occurred during Fracture under Conditions of Ductile-Brittle Transition Temperature Region

It has been shown by means of EBSD techique that fracture of ferritic steel in ductile-brittle transition temperature region, along with the formation of previously discribed cleavage microcracks, results in the formation of ductile microcracks. It has also been shown that microstructure of plastic zones under brittle and ductile fracture components produced by the main crack propagation differ significantly. Better developed plastic zone under ductile fracture component protects steel from overstress. The plastic zone under brittle fracture surface, apparently, has a reduced local plasticity. Consequently, the cleavage microcracks formation precedes the fracture process. During the main crack formation such microcracks occur in steel microvolumes located both in front of its tip and in adjacent to its edges microvolumes. Further propagation of the main crack is realized in steel which already contains scattered cavities and reduces to ductile fracture of the connections between them.     

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.     

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.     

Computational modeling of crack propagation in real microstructures of steels and virtual testing of artificially designed materials

A computational approach to the optimization of service properties of two-phase materials (in this case, fracture resistance of tool steels) by varying their microstructure is developed. The main points of the optimization of steels are as follows: (1) numerical simulation of crack initiation and growth in real microstructures of materials with the use of the multiphase finite elements (MPFE) and the element elimination technique (EET), (2) simulation of crack growth in idealized quasi-real microstructures (net-like, band-like and random distributions of the primary carbides in the steels) and (3) the comparison of fracture resistances of different microstructures and (4) the development of recommendations to the improvement of the fracture toughness of steels. The fracture toughness and the fractal dimension of a fracture surface are determined numerically for each microstructure. It is shown that the fracture resistance of the steels with finer microstructures is sufficiently higher than that for coarse microstructures. Three main mechanisms of increasing fracture toughness of steels by varying the carbide distribution are identified: crack deflection by carbide layers perpendicular to the initial crack direction, crack growth along the network of carbides and crack branching caused by damage initiation at random sites.     

Scaling properties of mortar fracture surfaces

Scaling properties of mortar crack surfaces are studied from mode I fracture specimens of six different sizes. Fracture surfaces initiated from a straight notch exhibit an anomalous dynamic scaling which involves two independent roughness indices: the universal local roughness exponent ζ_loc ≈ 0.8 and the global roughness exponent, estimated to ζ ≃ 1.35. We show that there exists a linear relationship between the specimen size and the maximum self-affine correlation length inducing a size effect on the roughness magnitude at saturation and this especially for the smallest length scales. Finally, we argue that anomalous roughening could be an inheritance of the changes in long range elastic interactions which take place in the fracture process zone of quasibrittle materials.     

Scale-effects on mean and standard deviation of the mechanical properties of condensed matter: an energy-based unified approach

The size effects on the mean values of the mechanical properties of condensed matter and on the related variances are analysed by means of a unified approach based on the multiscale character of energy dissipation. In particular, the scaling law for fragmentation energy density is obtained taking into account the self-similarity of fragments. It is based on a generalization of the three classical comminution laws that has been performed to evaluate the energy dissipation, computing volume and surface area of the particles for one- two- and three-dimensional fragmented objects. The result is general and can be applied to different fractal energy dissipation mechanisms, e.g., plasticity. Based on this approach, the scaling laws for mean and standard deviation values of the main mechanical properties of materials can be derived, like Young's and shear elastic moduli, ultimate normal and shear stresses and strains, fracture energy and toughness.     

Scale-effects on mean and standard deviation of the mechanical properties of condensed matter: an energy-based unified approach

The size effects on the mean values of the mechanical properties of condensed matter and on the related variances are analysed by means of a unified approach based on the multiscale character of energy dissipation. In particular, the scaling law for fragmentation energy density is obtained taking into account the self-similarity of fragments. It is based on a generalization of the three classical comminution laws that has been performed to evaluate the energy dissipation, computing volume and surface area of the particles for one- two- and three-dimensional fragmented objects. The result is general and can be applied to different fractal energy dissipation mechanisms, e.g., plasticity. Based on this approach, the scaling laws for mean and standard deviation values of the main mechanical properties of materials can be derived, like Young's and shear elastic moduli, ultimate normal and shear stresses and strains, fracture energy and toughness.     

Micromechanics of fracture of solid polymers in dynamic loading

The results of investigation of the influence of temperature on the micromechanics of fracture ofpofymethyl methacrylate and epoxy materials in dynamic loading are presented, A method was developed for determination of the value characterizing the formation and development of microcracks in front of the front of the main crack. Anomalous temperature relationships of crack length and the density of microfractures in polymethyl methacrylate were established. The influence of the processes of stress concentration, local inelastic deformation, and thermofluctuation rupture of polymer chains on the micromechanics of fracture of polymers is considered. It is shown that a change in temperature may lead to a qualitative change in the processes of microfractures. The formation of microcracks in front of the front of a propagating crack may transform into outstripping of it by the leading microcracks and the reverse. It is established that a thermofluctuation character is characteristic of fracture of polymers in intense short-term loading.Translated from Problemy Prochnosti, No. 2, pp. 80–85, February, 1990.     

Quasibrittle fracture beneath a flat bearing surface

The problem of fracture of the quasibrittle and brittle material such as glass when subject to indentation by a relatively rigid, square-ended uniform punch is considered, using classical energy principles. The strain energy needed to form a crack adjacent to the punch edge is found, and from this the load needed to initiate fracture given. The calculation is of practical interest in the design rating of bearing blocks supporting structural glass. Additionally, it implies also the potential possibility to set up a very simple and practical technique for evaluating some strength-related properties of materials such as the fracture toughness using specimens without macro-pre-existing cracks.     

Critical linkages of coalescing microcracks under stress loading

ABSTRACT The fracture process of brittle materials with randomly orientated microcracks critically depends on strong interactions among microcracks and the coalescence path that leads to a fatal crack. In this paper, a model based on the coalescence process for planar orientated microcracks is presented. An energy ratio is defined as the competition between the potential energy release and the new crack surface energy in each coalescence step, which is a token of the excessive driving force for microcrack propagation. A critical linkage dictates the coalescence of microcracks under stress loading. Probabilities of microcrack coalescence dominated by the first linkage and subsequent linkages are analysed for collinear and wavy microcrack arrays in detail.     

Fracture behaviour of liquid crystal epoxy resin systems based on the diglycidyl ether of 4,4′-dihydroxy-α-methylstilbene and sulphanilamide: Part I Effects of curing variations

The fracture behaviours of the pour-cast, unoriented diglycidyl ether of 4,4′-dihydroxy-α-methylstilbene/sulphanilamide liquid crystalline epoxies (LCE) cured at various temperature steps are investigated. It is found that, depending on how the LCE is cured, the liquid crystalline (LC) domain size varies dramatically. These, in turn, affect how the LCEs fracture. The operative toughening mechanisms in the toughest LCE are studied in detail and found to include the formation of numerous segmented, unlinked microcracks in front of the main crack. When the crack opens up, the matrix material between the segmented microcracks acts as a bridge between the opening crack planes. Furthermore, crack bifurcation appears to take place when the segmented cracks are eventually linked with the main crack. This entire fracture process accounts for the high fracture toughness (GIC=580 J m-2) of this particular LCE with respect to conventional epoxies (GIC=180 J m-2). The relationship between the LCE morphology and the corresponding fracture mechanisms is discussed. This revised version was published online in November 2006 with corrections to the Cover Date.     

Nano-mechanics based modeling of lifetime distribution of quasibrittle structures

The statistics of structural lifetime under constant load are related to the statistics of structural strength. The safety factors applied to structural strength must ensure failure probability no larger than 10^-6, which is beyond the means of direct verification by histogram testing. For perfectly brittle materials, extrapolation from the mean and variance to such a small tail probability is no problem because it is known that the Weibull distribution applies. Unfortunately, this is not possible for quasibrittle materials because the type of cumulative distribution function (cdf) has been shown to vary with structure size and shape. These are materials with inhomogeneities and fracture process zones (FPZ) that are not negligible compared to structural dimensions. A probabilistic theory of strength of quasibrittle structures failing at macro-crack initiation, which can be experimentally verified and calibrated indirectly, has recently been deduced from the rate of jumps of atomic lattice cracks governed by activation energy barriers. This paper extends this nano-mechanics based theory to the distribution of structural lifetime. Based on the cdf of strength and a power law for subcritical crack growth rate, the lifetime cdf of quasibrittle structures under constant loads is derived. The lifetime cdf is shown to depend strongly on the structure size as well as geometry. It is found that, for the creep rupture case, the mean structural lifetime exhibits a very strong size effect, much stronger than the size effect on the mean structure strength. The theory also implies temperature dependence of the lifetime cdf. For various quasibrittle materials, such as industrial ceramics and fiber composites, it is demonstrated that the proposed theory correctly predicts the experimentally observed deviations of lifetime histograms from the Weibull distribution.