Fractal Crack Research Articles

Strength evalution of retrofit shear wall elements with interfaces of fractal geometry

In the present paper structures involving rough interfaces, with irregularities of all scales, are simulated with the help of fractal geometry. More specifically, the approach followed is the multilevel hierarchical approach. This approach has a multiscale self-affine structure appropriate for the simulation of rough interfaces. In the sequence the problems resulting from the consideration of a fractal interface as a profile with hierarchical structure are analyzed, with the assumption that on the interface nonmonotone adhesive contact and friction conditions hold. The method developed consists of an extension of the classical FEM to the case of fractal interfaces. The case of nonconvex energy problems is investigated, and some results from static analysis of concrete shear walls with prescribed fractal cracks are included in order to illustrate the theory.

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.

On Fractal Cracks in Micropolar Elastic Solids

In this paper we review the fracture mechanics of smooth cracks in micropolar (Cosserat) elastic solids. Griffith’s fracture theory is generalized for cracks in micropolar solids and shown to have two possible forms. The effect of fractality of fracture surfaces on the powers of stress and couple-stress singularity is studied. We obtain the orders of stress and couple-stress singularities at the tip of a fractal crack in a micropolar solid using dimensional analysis and an asymptotic method that we call “method of crack-effect zone.” It is shown that orders of stress and couple-stress singularities are equal to the order of stress singularity at the tip of the same fractal crack in a classical solid.

Fractal Fracture of an Elastic Plane Weakened by a Lunate Notch

The possibility of satisfying the energy balance conditions is justified for the problem of tension on a plane weakened by a notch in the shape of a symmetric lune. A macrocrack growing from a corner of an elastic domain is modeled by a fractional-dimensional fractal. The dependence of the dimension of a fractal crack on the angle of the lunate notch is determined such that the Griffith energy balance equation can be satisfied. The same kind of determination is impossible in the realm of classical fracture mechanics and is used as a basis for calculating the breaking load in the stated problem.

Percolation in the crust derived from distortion of electric fields

Fractal crack distributions in the crust have been discussed in the context of earthquake cycles, fluid percolation, and the electrical conduction mechanism. With the impedance matrix rotation analysis in magnetotellurics (MT), a special type of apparent amplification of variational electric fields in one horizontal direction was detected. Here a possible link between crustal fracture networks and MT distortion is suggested. It is shown that the strong electric field amplification can be explained with a very heterogeneous crack network which in one direction stays in the vicinity of the percolation threshold. The apparent similarity to fluid percolation models does not automatically support the fluids paradigm, but rather allows for a choice between present fluid percolation and ‘paleopercolation’ to be the origin of the conductivity structure. In the latter model, the fluid caused the precipitation of graphite which now forms a partly interconnected network.

The fourth mode of fracture in fractal fracture mechanics

This paper offers a systematic approach for obtaining the order of stress singularity for different self-similar and self-affine fractal cracks. Mode II and Mode III fractal cracks are studied and are shown to introduce the same order of stress singularity as Mode I fractal cracks do. In addition to these three classical modes, a Mode IV is discovered, which is a consequence of the fractal fracture. It is shown that, for this mode, stress has a weaker singularity than it does in the classical modes of fracture when self-affine fractal cracks are considered, and stress has the same order of singularity when self-similar cracks are considered. Considering this new mode of fracture, some single-mode problems of classical fracture mechanics could be mixed-mode problems in fractal fracture mechanics. By imposing a continuous transition from fractal to classical stress and displacement fields, the complete forms of the stress and displacement fields around the tip of a fractal crack are found. Then a universal relationship between fractal and classical stress intensity factors is derived. It is demonstrated that for a Mode IV fractal crack, only one of the stress components is singular; the other stress components are identically zero. Finally, stress singularity for three-dimensional bodies with self-affine fractal cracks is studied. As in the two-dimensional case, the fourth mode of fracture introduces a weaker stress singularity for self-affine fractal cracks than classical modes of fracture do.

Fractal evolution of a crack network in overburden rock rtratum

A quantitative description for the spatial distribution and the evolution of a crack network in mining rock stratum is one of the most difficult and fundamental problem in the subject of surface subsidence. In this paper, the physical models are employed to simulate the spatial distribution of a crack network. By using the fractal geometry, the self‐similarity of spatial distribution of crack network is discovered. As a result, the conception of fractal crack network is proposed. Furthermore, the evolution of a crack network with the increasing of mining width is investigated. It is shown that (1) the spatial distribution of a crack network displays the fractal behavior, so, the fractal dimension can be used to describe quantitatively the evolution of the crack network, (2) the fractal dimension of the crack network increases with increasing of mining width, (3) the surface subsidence increases with the increasing of fractal dimension of crack network.

Acoustic characterization of fractal crack surface

Evaluation of fractal dimensions of crack surfaces in materials has now become essential in order to correlate the properties of the micro structure responsible for the evolution of the crack. A model has been developed which estimates the Hurst exponent (the parameter characterizing the fractal nature) of the crack surface from the observed scattered intensity of a quasi-monochromatic acoustic pulse incident on it.

Evolution of microseismicity during frictional sliding

We have done frictional sliding experiments on Inada granite in double shear and monitored the acoustic emission (AE) produced and temporal changes in the microseismic b‐value (where b is defined as the log‐linear slope of the AE frequency‐amplitude distribution), using both rough and smooth ground simulated fault surfaces. We have found, (i) the maximum amplitudes of AE events during stable sliding are strongly dependent on the surface roughness with smooth‐ground surfaces giving smaller maximum AE amplitudes; (ii) b‐values are related to the surface topographic fractal dimensions, so that in steady‐state stable sliding smooth surfaces exhibit lower b‐values than rough surfaces; (iii) the b‐value falls before stick‐slip instability. The change of b with slip we interpret in terms of evolving fractal crack damage during frictional sliding of the fault surfaces.

On Debonding and Delamination Effects in Adhesively Bonded Cracks on Fractal Type

The present paper studies the influence of adhesives on the behaviour of cracks in deformable bodies. Especially the delamination and debonding effects are studied. The adhesive material is assumed to introduce nonmonotone possibly multivalued laws which can be described via nonconvex superpotentials. Cracks of fractal geometry are considered. Approximating the fractal crack by a sequence of smooth surfaces or curves and combining this procedure with an algorithm based on the optimization of the potential and of the complementary energy for each approximation of the fractal crack, we get the solution of the problem. Numerical examples, using singular elements for the consideration of the crack singularity, illustrate the theory.

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 , on the crack length, and on the material length 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.

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.

Fractal effects of crack propagation on dynamic stress intensity factors and crack velocities

Crack extension paths are often irregular, producing rough fracture surfaces which have a fractal geometry. In this paper, crack tip motion along a fractal crack trace is analysed. A fractal kinking model of the crack extension path is established to describe irregular crack growth. A formula is derived to describe the effects of fractal crack propagation on the dynamic stress intensity factor and on crack velocity. The ratio of the dynamic stress intensity factor to the applied stress intensity factor K(L(D, t), V)/K(L(t), 0), is a function of apparent crack velocity Vo, microstructure parameter d/Δa (grain size/crack increment step length), fractal dimension D, and fractal kinking angle of crack extension path ϑ. For fractal crack propagation, the apparent (or measured) crack velocity Vo, cannot approach the Rayleigh wave speed Cr. Why Vo is significantly lower than Cr in dynamic fracture experiments can be explained by the effects of fractal crack propagation. The dynamic stress intensity factor and apparent crack velocity are strongly affected by the microstructure parameter (d/Δa), fractal dimension D, and fractal kinking angle of crack extension path ϑ. This is in good agreement with experimental findings.

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.

Scaling of Quasi-Brittle Fracture and the Fractal Question

The paper represents an extended text of a lecture presenting a review of recent results on scaling of failure in structures made of quasibrittle materials, characterized by a large fracture process zone, and examining the question of possible role of the fractal nature of crack surfaces in the scaling. The problem of scaling is approached through dimensional analysis, the laws of thermodynamics and asymptotic matching. Large-size and small-size asymptotic expansions of the size effect on the nominal strength of structures are given, for specimens with large notches (or traction-free cracks) as well as zero notches, and simple size effect formulas matching the required asymptotic properties are reported. The asymptotic analysis is carried out, in general, for fractal cracks, and the practically important case ofnonfractal crack propagation is acquired as a special case. Regarding the fractal nature of crack surfaces in quasibrittle materials, the conclusion is that it cannot play a signification role in fracture propagation and the observed size effect. The reason why Weibull statistical theory of random material strength does not explain the size effect in quasibrittle failures is explained. Finally, some recent applications to fracture simulation by particle models (discrete element method) and to the determination of size effect and fracture characteristics of carbon-epoxy composite laminates are briefly reviewed.

Effects of fractal crack

Experimental results indicate that propagation paths of cracks in geomaterials are often irregular, producing rough fracture surfaces which are fractal. In this paper, crack tip motion along a fractal crack trace is discussed. A fractal kinking model of the crack extension path is established to describe irregular crack growth. The length, velocity and kinking effects of the fractal crack are analysed. A formula is derived to describe the effects of fractal crack propagation on the dynamic stress intensity factor and on crack velocity. Finally, expressions of stress and displacement fields near the fractal crack tip are given.

Fractal kinematics of crack propagation in geomaterials

Experimental results indicate that propagation paths of cracks in geomaterials are often irregular, producing rough fracture surfaces which are fractal. A formula is derived for the fractal kinematics of crack propagation in geomaterials. The formula correlates the dynamic and static fracture toughness with crack velocity, crack length and a microstructural parameter, and allows the fractal dimension to be obtained. From the equations for estimating crack velocity and fractal dimension it can be shown that the measured crack velocity ( V 0) should be much smaller than the fractal crack velocity ( V). It can also be shown that the fractal dimension of the crack propagation path can be calculated directly from V 0 and from the fracture toughness.

EXAMPLES OF FRACTALS IN SOIL MECHANICS

Models will be presented for fractal structures appearing naturally in soils. On the one hand, we discuss the opening of brittle media via hydraulic fracturing at constant pressure using a square lattice beam model with disorder. We consider the case in which only beams under tension can break, and discuss under which conditions the resulting cracks may develop fractal patterns. The stress field of the fractal cracks is visualized by photoelastic fringes. Then we present a modelization for a fluid penetrating under a pressure gradient into a fractal crack which it is itself opening. To do this, we investigate invasion percolation fingers in a quenched medium in which the randomness has a gradient corresponding to the density of microcracks that arise in a self-organized way around a large crack.

Models of fractal cracks.

We present theoretical ideas which allow us to understand part of the scaling laws recently observed on branched cracks. We argue that some features are common to all critical branched structures, which barely survive when propagating. These ideas are illustrated by the directed percolation problem, which serves as an excellent toy model. Finally, we propose a Langevin equation for unbranched cracks in three dimensions, which naturally leads to self-affine structures.

Fractal cracks

An attempt is made to take into account the irregular structure of the surface of a real crack in describing the fracture process. The crack surface is modelled using a fractal set of fractional dimension. Using the self-similarity of the fractal, a hierarchical process of the transfer of elastic energy generated during the motion of the crack tip from one scale to another is suggested. Analysis of this process makes it possible to obtain asymptotic expressions for describing the behaviour of the cracks and displacements near the crack tip. It is shown that the fractal geometry of the crack leads to a change in the singular behaviour of the stress fields at the crack tip, and to the appearance of an anomalous dimensionally dependent factor in the expression for the stress intensity factor. Similar results are also obtained for branching fractal cracks. The propagation of a fractal crack in a brittle material is analysed from the positions of the Griffith's criterion.