Jtam-A4.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 52, 1, pp. 37-46, Warsaw 2014 UBIQUITIFORM IN APPLIED MECHANICS Zhuo-Cheng Ou, Guan-Ying Li, Zhuo-Ping Duan, Feng-Lei Huang Beijing Institute of Technology, State Key Laboratory of Explosion Science and Technology, Beijing, China e-mail address: zcou@bit.edu.cn Wedemonstrate that a physical object innature should notbe described as a fractal, despite an ideal mathematical object, rather a ubiquitiform (a terminology coined here for a finite order self-similar or self-affine structure). It is shown mathematically that a ubiquitiform must be of integral dimension, and that the Hausdorff dimension of the initial element of a fractal changes abruptly at the point at infinity, which results in divergence of the integral dimensional measure of the fractal and makes the fractal approximation to a ubiquitiform unreasonable. Therefore, instead of the existing fractal theory in applied mechanics, a new type of ubiquitiformal one is needed. Key words: ubiquitiform, fractal, Hausdorff dimension 1. Introduction A fractal (Mandelbrot, 1967, 1977, 1982) or, generally, a self-similar (or self-affine) structure of infinite order, has been extensively used as a nonlinear mathematical tool to describe various irregular and complex phenomena in many fields of science and technology such as fracture mechanics (Mandelbrot et al., 1984; SaoumaandBarton, 1994;Carpinteri, 1994;Borodich, 1997, 1999; Carpinteri and Paggi, 2010), quantum mechanics (Argyris et al., 2000), material science (Fleury, 1997; Ma et al., 2009), nonlinear dynamics and chaos (Aguirre et al., 2009), biology (Gözen et al., 2010) and even city planning (Batty, 2008). In fracture mechanics, for example, sinceMandelbrot’s pioneeringwork(1984) inwhich the fractality of an impacting fracture surface in metal was found, fractal fracture mechanics has been developed quickly in several aspects. For example, on the one hand, the fractality of fracture surfaces in various kinds of materials such as steel (Mandelbrot et al., 1984), concrete (Saouma et al., 1990; Saouma and Barton, 1994), ceramic (Mecholsky et al., 1989) and rock (Krohn and Thompson, 1986) was confirmed by experimentalmethods. On the other hand, it was found that the distribution ofmicro-cracks and micro-damage could be described by fractals (Barenblatt and Botvina, 1986; Chudnovsky and Wu, 1992; Barenblatt, 1993). Moreover, the fractal geometry was also used to describe the mechanical behavior of materials such as the size effect on material strength (Carpinteri, 1994; Carpinteri and Puzzi, 2008; Carpinteri and Paggi, 2010) and crack propagation (Xie and Sanderson, 1995), and, at the same time, the fracture problem in regionswith fractal boundaries also became a research focus (Panagiotopoilos et al., 1995; Wnuk and Yavari, 2005). With the development of fractal application, some inherent difficulties appear gradually because that a fractal is just an ideal mathematical object with a self-similar (or self-affine) structure of infinite hierarchy. For example, in fractal fracture mechanics, some authors have endeavored to link the fractal dimension of the fracture surface to fracture toughness, but a general conclusion cannot be drawn easily. Actually, all three kinds of the correlation, namely, the negative correlations (Mandelbrot et al., 1984; Saouma and Barton, 1994), the positive correlations (Mecholsky et al., 1989; Ray and Mandal, 1992) as well as no correlation (Pande et al., 1987) can be found. It is not hard to imagine that such difficulties should be resulted from the uncertainty in describing a fractal. That is, a fractal could not be uniquely determined 38 Z.-C. Ou et al. just by its fractal dimension. In fact, as is well known, besides the dimension, in general, a finite measure is another significant parameter in describing a geometric object, such as that all the line segments are of one-dimension, but they can have different length or measure and hence two line segments with different lengths are always not equivalent in practice. However, it is difficult to calculate the measure of a fractal, let alone that some fractals even have no finitemeasure at all.Moreover, the integral dimensionalmeasure of a fractal is usually divergent or singular, namely, the integral dimensional measure of a fractal is either infinite or zero. In addition, traditional physical parameters are always defined in integral dimensional measure, but the direct extension of these parameters in fractal application seems not available andmay cause unreasonable results. For example, the fracture energy is defined as the work done by the external force to generate aunit fracture surface (smooth or in integral dimensionof D=2), but for a fractal fracture surfacewith the fractal dimension D> 2, the corresponding area is infinite because of the divergence of the integral dimensionalmeasure of the fractal surface. This implies that the infinite amount of fracture energy will be needed for the fracture of a material, which is certainly not the case. To accommodate the integral dimensional divergence of a fractal, new density kinds of fractal parameters defined on the unit fractal measure had to be introduced, such as the specific energy-absorbing capacity of unit fractal measure (Borodich, 1997, 1999), the fractal tensile strength and the critical fractal strain (Carpinteri, 1994; Carpinteri et al., 2002), but these fractal parameters are both difficult to be determined in practice and seem lacking of unambiguous physical meanings (Baz̆ant and Yavari, 2005). Moreover, it should also be mentioned that, in general, the boundary-value problem in a region with fractal boundaries is difficult to be constructed because the normal direction of the fractal boundary cannot be defineddue to that amathematical fractal curve can be continuous everywhere butdifferentiable nowhere.Although somehomogenizationmethodswere usedbyprevious researchers (Davey and Alonso Rasgado, 2011; Davey and Prosser, 2013), but these methods are usually approximate and lacking of theoretical foundation. In fact, from amathematical point of view, a fractal is an infinite set with a self-similar (or self-affine) structure of infinitehierarchy,which canusuallybeconstructedbyan infinite iterative procedure. On the contrary, however, a physical object in nature can never experience such an infinite iterative process but only a finite one, and thus has a lower bound of the scale (such as the scale of the elementary particle or the time limit of a physical process). Two questions arise then: Is aphysical object in nature really a fractal? If not, is it reasonable or available to describe a natural object approximately by a fractal? It was believed that there is not any real fractal in nature (Mandelbrot, 1977; Falconer, 2003), but a physical object in nature could be described as or approximated by a fractal, just as presented byMandelbrot (1977), “Intuitively as well as pragmatically (from the point of view of the simplicity and naturalness of the corrective terms required), it is reasonable to consider a very close approximation to the Koch curve as closer to a curve of dimension log4/ log3 than to a curve of dimension 1”, However, as will be shown later, it is not the case but antipodal. That is to say, such a close approximation to the Koch curve with a finite iterative procedure must be a curve of integral dimension 1 rather than a curve of fractional dimension log4/ log3. Actually, it has been noticed that a physical object in nature could be described as or approximated by a fractal only in a finite range of scaling invariance in all fractal applications (Mandelbrot, 1982; Cherepanov et al., 1995; Balankin, 1997; Borodich, 1999; Addison, 2000; Martinez-Lopez et al., 2001). Moreover, we will show that the fractal approximation of a physical object is unreasonable especially in the sense of measure for the divergence of the integral dimensional measure or the discontinuity of the Hausdorff dimension of a fractal. In order to avoid the weakness of fractals used in applied mechanics as mentioned above, a new concept of ubiquitiform, a newword (with its adjective form “ubiquitiformal”) coined here from the abbreviated word combination of “ubiquitous” and “form” to imply the ubiquitous Ubiquitiform in applied mechanics 39 physical forms, is introduced in this paper.Aubiquitiform is defined as a finite order self-similar (or self-affine) physical configuration constructedusuallyby afinite iterative procedure, and thus it has the same Hausdorff dimension of the initial element which is always integral in practice. Moreover, a ubiquitiform is also different with a “prefractal” which was used by some previous authors for “a fractal in a finite range of scaling invariance” (Borodich, 1997; Addison, 2000). Although a prefractal is also produced by a finite iterative procedure, but by definition, it is also regarded as a fractal such as it has the same fractal dimension as that of the corresponding fractal, etc. However, a ubiquitiformal curve is just one-dimensional. Additionally, for the sake of simplicity, just the self-similar case is considered in the following. 2. Ubiquitiform or fractal? Consider in the two-dimensional Euclidean space a fractal curve with the length of the initial element l and fractal dimension D (1D (2.4) 40 Z.-C. Ou et al. where {Ui}and D are the δ-cover and theHausdorffdimensionof F, respectively, and |•|deno- tes a certainEuclideanmetric. As iswell known, a self-similar structure can usually be expressed as a union of finite or infinite (corresponding to a ubiquitiformand fractal, respectively) number of subsets resulted from a series of repetitious contractions starting from an initial element, say F0 of integral dimension d, with the finite d-dimensional Hausdorffmeasure, namely H(d)(F0)= lim δ→0 inf ∞ ∑ i=1 |Ui| d <∞ (2.5) where {Ui} is the δ-cover of F0.Without loss of generality, assume that the ubiquitiform Fn is generated by n steps of contractions and can be mapped into the corresponding subsets Fk n+1 (k = 1, . . . ,m) of the ubiquitiform Fn+1 generated by n+1 steps of contractions with the scaling λk < 1 (k=1, . . . ,m), where m is the number of contractions in the iterated function system (IFS). According to induction, supposing that Fn is of dimension d, i.e. H(d)(Fn)= lim δ→0 inf ∞ ∑ i=1 |Vi| d <∞ (2.6) where {Vi} is the δ-cover of Fn, and if δk = λkδ, then {λkVi} will be the δ-cover of F k n+1 (k = 1, . . . ,m). According to the scaling properties of the Hausdorff measure (Falconer, 2003), one can obtain H(d)(Fk n+1)=λ d k H(d)(Fn)¬λ d kmax H(d)(Fn) (2.7) where λkmax is the maximum of λk (k=1, . . . ,m). For Fn+1 = m ⋃ k=1 Fk n+1 (2.8) there is H(d)(Fn+1)=H (d) ( m ⋃ k=1 Fk n+1 ) ¬ m ∑ k=1 H(d)(Fk n+1)¬mλ d kmax H(d)(Fn) (2.9) Considering that mλd kmax is a finite value, form Eqs. (2.6) and (2.9), one obtains H(d)(Fn+1)<∞ (2.10) which means that Fn+1 has the finite d-dimensional Hausdorff measure. This implies that the Hausdorff dimension of Fn+1 is d by considering the uniqueness of the Hausdorff dimension. Therefore, a given ubiquitiformmust be of integral dimension. The above proof has also revealed an important fact that theHausdorff dimension is discon- tinuous at the point at infinity of δ → 0 in the iterative procedure of a fractal, or, in other words, the Hausdorff dimension of a fractal changes abruptly in the point of δ = 0, and the corresponding configuration changes from a ubiquitiformwith an integral dimension to a fractal with a fractal dimension. TheHausdorff dimension remains all through an integer for δ > 0 and hence the configuration is a ubiquitiform, but it jumps to a fractional value just when δ → 0, and the configuration becomes a fractal. Thus, it can be easily understood that a fractal with an infinite iteration processmay have a fractional Hausdorff dimension, but “a fractal in a finite range of scaling invariance” must be of an integral Hausdorff dimension, which can then be de- scribed only by a ubiquitiform. Taking the Koch curve as an example (see Fig. 1), its Hausdorff dimension remains to be 1 for arbitrary finite times of iteration k, which can be called as a Ubiquitiform in applied mechanics 41 Fig. 1. Hausdorff dimension of the ubiquitiformal and the fractal Koch curve ubiquitiformal Koch curve, and jumps from 1 to 1.26 at the infinite point of k →∞, which is rigorously the fractal Koch curve. As mentioned above, a physical object in nature is a ubiquitiform. In such a case, is it reasonable to describe a ubiquitiform as a fractal? We will show that such an approximation is unavailable in the sense of the Hausdorff measure (in most cases of fractal application). Comparison Eq. (2.2) with Eq. (2.3) shows that the difference in the one-dimensional measure between a ubiquitiformal and its associated fractal curve is infinitely large, which makes the fractal approximation to a ubiquitiform unreasonable. The basic reason is that the Hausdorff dimension of a ubiquitiform is different from that of its associated fractal.Moreover, it is worthy of special mention that Panagiotopoulos et al. (1993) and Falconer (2003) have argued that one can obtain very good approximations of a fractal by an iterated function system constructed from a given set in the sense of the Hausdorff distance. However, on the one hand, such an iterative process is after all finite and then could not realize the approximation of the Hausdorff dimension because of the discontinuity of the Hausdorff dimension at the infinite point with respect to the Hausdorff distance, or, in other words, the approximation in the sense of the Hausdorff distance does not imply that in the Hausdorff dimension. On the other hand, it must be noticed that a physical object in nature is the objective reality, and, from the viewpoint of scientific research,whatweneed is touse the fractal as anavailablemathematical tool todescribe such objectivity, but not vice versa. Unfortunately, although theremay exist some set sequences approximating to a certain fractal within a satisfactory precision, the inverse is not true.That is, a given physical object in nature can never bewell approximated by any fractal especially in the sense of measure. It is also worthy of mention that some similar approximations seem existing in practice, such as the mass density in continuum mechanics, which is definedmathematically in the limit case of the mass-to-volume ratio ∆M/∆V when the small volume ∆V shrinks to zero, but physically, ∆V is still kept above some finite value ε > 0 larger compared with the mean free path of themolecules. Such an ideal of “macro-infinitesimal andmicro-infinite” is the foundation of continuummechanics. Itmust benoticed that themass density can always tend to a stable value when the small volume shrinks, say, to a finite value ε, and hence the density can be definedwell. However, for a fractal, its integral dimensionalmeasure is always divergent with thedecreasingmeasure scale andcannot reacha stable value. Inaddition, it hasalsobeennoticed that the stress lim∆S→0∆F/∆S (∆S and ∆F is a small area and the resultant surface force acting on ∆S respectively) has played an important role inmacro continuummechanics, but in micromechanics such as nanomechanics, researchers have paid attention to some new integrated parameter such as energy-momentum tensor and configurational force, because the divergence of such a limit has causedmore andmore difficulties. In summary, it is unreasonable to describe a physical object in nature as a fractal because of the divergence of the integral dimensional measure of the fractal. From this perspective, the real physical object in nature which had 42 Z.-C. Ou et al. ever been approximated by a fractal should be described as a ubiquitiform, or further, the real physical object in nature is a ubiquitiform. In fact, the splendid “fractal pictures” drawn on a piece of paper can only be two-dimensional, and thePiano curve of a finite order is also not able to cover any finite area (this is imaginable considering that a curve is of zero-width), which is just a one-dimensional ubiquitiformal curve. 3. Discussion As described above, we emphasize again that the geometry of nature is ubiquitiformal rather than fractal, and any physical object in nature should not be described as a fractal, rather a ubiquitiform, which is of integral dimension in the Euclidean space. There is inherent difference between a fractal and a ubiquitiform, especially in the case of the Hausdorff dimension, which has been ignored for a long time. Comparing with a fractal, one of the most important cha- racterizations of a ubiquitiform is that it has a finite integral dimensional measure. Equation (2.3) shows that the one-dimensional measure of a ubiquitiformal curve depends not only on the measure of the initial element but also on the complexity D and the lower bound to the scale invariance δmin. As has been shown above, the complexity of a ubiquitiform is equal to the fractal dimension of the corresponding fractal, and this indicates that achievements obtained in fractal application will play an important role in the ubiquitiformal theory. It can be seen from Eq. (2.3) that the lower bound to the scale invariance δmin is a crucial parameter for a ubiquitiform,which is usually assumed to beamaterial constant. However, until now, the understanding of this parameter is far from complete. The key points are which factor will affect the lower bound to the scale invariance δmin andhow todetermine its value accurately for a certain material. For a certain ubiquitiform with unknown δmin, it seems that δmin can be obtained from Eq. (2.3) under a given iterative procedure, which can be termed as the computational lower bound δminc. However, it should be pointed out that δminc depends on the measure of the initial element l aswell as the complexity D and thenmay not be equal to δmin. For clarity, as an example, numerical results are presented by taking the ubiquitiformal Koch curve. Assume that l = 1mm and δmin = 1µm, the yardstick length of the nth iteration can be obtained as δn = 1/3 nmm, and then the maximum number of the iteration nmax can be determinedby taking δn theminimumvalue greater than δmin, i.e., nmax = [3/ log3]= 6,where the square brackets represent themaximum integer less than the argument. The computational lower bound is then δminc =1/3 6mm∼=1.37µm.Moreover, taking l=10mm, it can be verified that nmax =8 and δminc ∼=1.52µm, which is about 10.9% greater than that for l=1mm and implies the dependence of the computational lower bound δminc on the length of the initial element l. On the other hand, the influence of the relative error of δmin on the measure of the ubiquitiform should also be noticed. For example, in the case of l = 1mm, although the relative error of δmin is 37%, it can be obtained fromEq. (2.3) that Lubif (δmin)∼=6.10mmand Lubif (δminc)∼=5.62mm with the relative error of 7.9%, much less than that of δmin, and such a relative error is acceptable in engineering application. Moreover, in general, it was believed that the dimension d of both the integral dimensional objects and fractals can be defined simply by (Turcotte, 1997) N(δ)δd =C (3.1) where N(δ) is the number of objects (i.e. fragments) with a characteristic linear dimension δ, C is a constant. However, as described above, all ubiquitiforms are of integral dimension such as d, but satisfy N(δ)δD =C δ­ δmin (3.2) Ubiquitiform in applied mechanics 43 where D is generally a fractional value which is obviously not the same as d, which is also the real justification that we call D as the “complexity” of the ubiquitiform. Therefore, it appears that Eq. (3.2) should be a sufficient rather than a necessary condition for the definition of the dimension of a object. Moreover, it is seen that the complexity D may be an important parameter to classify objects in integral dimensions. That is to say, the objects in the integral dimension d could be classified into two categories: one is that they satisfy Eq. (3.1), or D= d, including for example a rectilinear segment or a smooth circular arc; the other one is that they satisfy Eq. (3.2) with D 6= d, as ubiquitiforms. Such a difference between the objects in a common dimension but different complexities seems to have been ignored for a long time. On the other hand, it is worth tomention that a fractal can be constructed from initial ele- mentswith different dimensions because theHausdorff dimension of the fractal can change in an infinite iterative procedure. This is verymuch different from that for a ubiquitiform. TheHaus- dorff dimension of the ubiquitiform is equal to the Hausdorff dimension of its initial element, but it is usually not the case for the fractal. For example, a Cantor set can be constructed from either an initial unit line segment or an initial point (Edger, 2008). The Hausdorff dimension of the initial line element is 1 and that of the initial point element is 0, but the fractal Cantor sets constructed from both initial elements have a common Hausdorff dimension of ln2/ ln3. However, the ubiquitiformalCantor sets come from such different initial elements, i.e. the initial point element and the initial line element, have different Hausdorff dimensions of 0 and 1, re- spectively, and can never be equivalent to each other.Obviously, the inherent difference between a ubiquitiform and its associated fractal comes from the jump characteristic of the Hausdorff dimension in the infinite iteration process. Moreover, in this way, it can be easily understood that the concepts of the invasive and the lacunar fractals (Baz̆ant and Yavari, 1997) will be meaningless in the sense of ubiquitiform, because the Hausdorff dimension of the ubiquitiform is always equivalent to that of its initial element. Moreover, it should be pointed out that, based on the concept of ubiquitiform, theoretically, some intrinsic difficulties in applied fractal science may be avoided. For example, on the one hand, a ubiquitiform is measurable in integral dimension and hence no any ambiguous physi- cal variable (such as the specific energy-absorbing capacity (Borodich, 1997, 1999), the fractal tensile strength and the critical fractal strain (Carpinteri, 1994; Carpinteri et al., 2002) propo- sed in fractal fracture mechanics) defined on the unit fractal measure seems necessary. All the corresponding physical variables can now be defined normally on the unit measure of integral dimension and thus will have unambiguous physical meanings. On the other hand, the finite in- tegral measures of any two ubiquitiforms in a common dimension but different complexities are now comparable, while on the contrary, it ismeaningless to compare themeasures of two fractal curves with different fractal dimensions. It should be emphasized that waking up to this point is of significance in practical applications because most of scientific criteria can be established based on such a comparison. For example, the fractal fracture energy can not be considered as one of important parameters to characterize the material toughness. The two fracture energies of two specimens with fracture surfaces in different fractal dimensions will be expressed in diffe- rentmechanical units. Under the concept of ubiquitiform, such an embarrassmentwill disappear naturally. Moreover, it can be expected that the incoming universal ubiquitiformal theory will be more easy-to-use in practice than the existing fractal one, since it can be constructed on the base of “classical” mathematics in the Euclidean space. For example the arduous fractal boundary value problems are difficult to be dealt with because the normal direction of a fractal boundary can not be defined, but using the ubiquitiformal theory, the problems turn into com- mon boundary value problems defined on piecewise smooth boundaries and can be treated by traditional mathematical methods. Additionally, it can be expected that the concept of ubiquitiformwill play an important role in the development of the critical distance theory (Taylor, 2004, 2007, 2008) and the quanti- 44 Z.-C. Ou et al. zed (finite or discrete) fracture mechanics (Pugno and Ruoff, 2004) proposed recently by some authors, because of the characteristic finite number iterative procedure. Moreover, combining the ubiquitiformal theory with the concept of incubation time in dynamic failuremechanics will also be propitious for the investigation of strain rate effects on material strength, in which the discrete structural response characteristic has been noticed by some authors recently (Cotsovos and Pavlović, 2008a,b,c; Ou et al., 2010). 4. Conclusion To overcome the intrinsic weakness of fractals used in applied mechanics, a new concept of ubiquitiform is introduced and discussed in this paper, and some conclusions can be drawn out as follows. • A physical object or a geometric configuration in nature can never be constructed by an infinite iterative procedure. In applied mechanics, therefore, instead of a fractal, despite an ideal mathematical object, a new concept of ubiquitiform should be introduced for a finite order self-similar (or self-affine) structure in nature in an attempt to overcome the weakness of the fractal such as, especially, its divergence or singularity of the integral dimensional measure. • It is shownmathematically that a ubiquitiform constructed from an initial element whose Hausdorffdimension is integer byafinite self-similar iterative proceduremustbeof integral dimension, which implies theoretically that all real physical objects in nature could not be of fractional dimension and then are not fractals. It is also found that the Hausdorff dimension is discontinuous at the point at infinity in the infinite iterative procedure of a fractal, which results in the radical difference between the ubiquitiform and the fractal. • It is also demonstrated that a given physical object in nature can never be well appro- ximated by any fractal, especially in the sense of the Hausdorff measure, because the ubiquitiform always has an finite Hausdorff measure, whereas the corresponding fractal has not. • Based on the concept of ubiquitiform, some intrinsic difficulties of fractals in applied mechanics can be avoided, e.g. it will not be necessary to introduce any kinds of density of fractal parameters defined on the unit fractal measure, and that the so-called fractal boundary-value problem could be constructed expediently in the sense of ubiquitiform. Acknowledgements This work were supported by The National Natural Science Foundation of China under Grant 11221202 and also Foundation of State Key Laboratory of Explosion Science and Technology under contract YBKT 08-11. References 1. Addison P.S., 2000, The geometry of prefractal renormalization: Application to crack surface energies,Fractals, 8, 147-153 2. AguirreJ.,VianaR.L., SanjuánM.F., 2009,Fractal structures innonlineardynamics,Reviews of Modern Physics, 81, 333-386 3. Argyris J.,CiubotariuC.I.,WeingaertnerW.E., 2000,Fractal space signatures inquantum physics and cosmology? I. Space, time,matter, fields and gravitation,Chaos, Solitons and Fractals, 11, 1671-1719 Ubiquitiform in applied mechanics 45 4. BalankinA.S., 1997,Physics of fracture andmechanics of self-affine cracks,Engineering Fracture Mechanics, 57, 135-203 5. Barenblatt G.I., 1993, Micromechanics of fracture, Theoretical and Applied Mechanics 1992, 25-52 6. Barenblatt G.I., Botvina L.R., 1986, Similarity methods in the mechanics and physics of fracture,Materials Science, 22, 52-57 7. Batty M., 2008, The size, scale, and shape of cities, Science, 319, 769-771 8. Baz̆antZ.P.,YavariA., 1997,Scaling of quasibrittle fracture: hypotheses of invasiveand lacunar fractality, International Journal of Fracture, 83, 41-65 9. Baz̆antZ.P.,YavariA., 2005, Is the causeof size effectonstructural strength fractalor energetic- statistical?Engineering Fracture Mechanics, 72, 1-31 10. Borodich F.M., 1997, Some fractal models of fracture, Journal of the Mechanics and Physics of Solids, 45, 239-259 11. Borodich F.M., 1999, Fractals and fractal scaling in fracture mechanics, International Journal of Fracture, 95, 239-259 12. Carpinteri A., 1994, Fractal nature of material microstructure and size effects on apparent mechanical properties,Mechanics of Materials, 18, 89-101 13. CarpinteriA.,ChiaiaB.,Cornetti P., 2002,A scale-invariant cohesive crackmodel for quasi- brittle materials,Engineering Fracture Mechanics, 69, 207-217 14. Carpinteri A., PaggiM., 2010,Aunified fractal approach for the interpretationof the anomalo- us scaling laws in fatigue and comparisonwith existingmodels, International Journal of Fracture, 161, 41-52 15. CarpinteriA.,Puzzi S., 2008,Self-similarity in concrete fracture: size-scale effects andtransition between different collapsemechanisms, International Journal of Fracture, 154, 167-175 16. CherepanovG.P.,BalankinA.S., Ivanova,V.S., 1995,Fractal fracturemechanics?A review, Engineering Fracture Mechanics, 51, 997-1033 17. ChudnovskyA.,Wu S., 1992, Evaluation of energy release rate in the crack-microcrack interac- tion problem, International Journal of Solids and Structures, 29, 1699-1709 18. Cotsovos D.M., Pavlović M.N., 2008a, Numerical investigation of concrete subjected to com- pressive impact loading. Part 1: A fundamental explanation for the apparent strength gain at high loading rates,Computers and Structures, 86, 145-163 19. Cotsovos D.M., Pavlović M.N., 2008b, Numerical investigation of concrete subjected to com- pressive impact loading. Part 2: Parametric investigation of factors affecting behaviour at high loading rates,Computers and Structures, 86, 164-180 20. Cotsovos D.M., Pavlović M.N., 2008c, Numerical investigation of concrete subjected to high rates of uniaxial tensile loading, International Journal of Impact Engineering, 35, 319-335 21. Davey K., Alonso Rasgado M.T., 2011, Analytical solutions for vibrating fractal composite rods and beams,Applied Mathematical Modelling, 35, 1194-1209 22. Davey K., Prosser R., 2013, Analytical solutions for heat transfer on fractal and pre-fractal domains,Applied Mathematical Modelling, 37, 554-569 23. Edgar G., 2008,Measure, Topology, and Fractal Geometry, Springer, NewYork 24. Falconer K., 2003,Fractal Geometry, Wiley, Chichester 25. Fleury V., 1997, Branched fractal patterns in non-equilibrium electrochemical deposition from oscillatory nucleation and growth,Nature, 390, 145-148 26. Gözen I., Dommersnes P., Czolkos I., Jesorka A., Lobovkina1 T., Orwar1 O., 2010, Fractal avalanche ruptures in biological membranes,Nature Materials, 9, 908-912 46 Z.-C. Ou et al. 27. Krohn C.E., Thompson, A.H., 1986, Fractal sandstone pores: Automated measurements using scanning-electron-microscope images,Physical Review B, 33, 6366-6374 28. Ma D., Stoica A.D., Wang X.L., 2009, Power-law scaling and fractal nature of medium-range order in metallic glasses,Nature Materials, 8, 30-34 29. MandelbrotB.B., 1967,How long is the coastofBritain?Statistical self-similarityand fractional dimension, Science, 156, 636-638 30. Mandelbrot B.B., 1977,Fractals: Form, Chance, and Dimension, Freeman, San Francisco 31. Mandelbrot B.B., 1982,The Fractal Geometry of Nature, Freeman, NewYork 32. Mandelbrot B.B., Passoja D.E., Paullay A.J., 1984, Fractal character of fracture surfaces of metals,Nature, 308, 721-722 33. Martinez-Lopez F., Cabrerizo-Vilchez M.A., Hidalgo-Alvarez R., 2001, An improved method to estimate the fractal dimension of physical fractals based on the Hausdorff definition, Physica A, 298, 387-399 34. Mecholsky J.J., PassojaD.E., Feinberg-Ringle K.S., 1989,Quantitative analysis of brittle fracture surfaces using fractal geometry, Journal of the American Ceramic Society, 72, 60-65 35. Ou Z.C., Duan Z.P., Huang F.L., 2010, Analytical approach to the strain rate effect on the dynamic tensile strength of brittle materials, International Journal of Impact Engineering, 37, 942-945 36. Panagiotopoulos P.D, Panagouli O.K., Koltsakis E.K., 1995, The B.E.M. in plane elastic bodies with cracks and/or boundaries of fractal geometry,Computational Mechanics, 15, 350-363 37. PanagiotopoulosP.D,PanagouliO.K.,Mistakidis E.S., 1993,Fractal geometryand fractal material behavior in solids and structures,Archive of Applied Mechanics, 63, 1-24 38. PandeC.S.,RichardsL.R.,SmithS., 1987,Fractal characteristicsof fracturedsurfaces,Journal of Materials Science Letters, 6, 295-297 39. Pugno N.M., Ruoff R.S., 2004, Quantized fracture mechanics, Philisophical Magazine, 84, 2829-2845 40. Ray K.K., Mandal G., 1992, Study of correlation between fractal dimension and impact energy in a high strength low alloy steel,Acta Metallurgica et Materialia, 40, 463-469 41. Saouma V.E., Barton C.C., 1994, Fractals, Fractures, and size effects in concrete, Journal of Engineering Mechanics, 120, 835-854 42. Saouma V.E., Barton C.C., Gamaleldin, N.A., 1990, Fractal characterization of fracture surfaces in concrete,Engineering Fracture Mechanics, 35, 47-53 43. TaylorD., 2004,Predicting the fracture strength of ceramicmaterials using the theory of critical distances,Engineering Fracture Mechanics, 71, 2407-2416 44. Taylor D., 2007,The Theory of Critical Distances, Elsevier, Oxford 45. TaylorD., 2008,The theory of critical distances,Engineering FractureMechanics, 75, 1696-1705 46. Turcotte D.L., 1997,Fractals and Chaos in Geology and Geophysics, Cambridge, NewYork 47. Wnuk M.P., Yavari A., 2005, A correspondence principle for fractal and classical cracks,Engi- neering Fracture Mechanics, 72, 2744-2757 48. Xie H.P., Sanderson D.J., 1995, Fractal kinematics of crack propagation in geomaterials,En- gineering Fracture Mechanics, 50, 529-536 Manuscript received January 10, 2013; accepted for print May 12, 2013