Extremum and variational principles for elastic and inelastic media with fractal geometries

This paper further continues the recently begun extension of continuum mechanics and thermodynamics to fractal porous media which are specified by a mass (or spatial) fractal dimension D, a surface fractal dimension d, and a resolution lengthscale R. The focus is on pre-fractal media (i.e., those with lower and upper cut-offs) through a theory based on dimensional regularization, in which D is also the order of fractional integrals employed to state global balance laws. In effect, the global forms of governing equations may be cast in forms involving conventional (integer-order) integrals, while the local forms are expressed through partial differential equations with derivatives of integer order but containing coefficients involving D, d and R. Here we first generalize the principles of virtual work, virtual displacement and virtual stresses, which in turn allow us to extend the minimum energy theorems of elasticity theory. Next, we generalize the extremum principles of elasto-plastic and rigid-plastic bodies. In all the cases, the derived relations depend explicitly on D, d and R, and, upon setting D = 3 and d = 2, they reduce to conventional forms of governing equations for continuous media with Euclidean geometries.     

Fractal dimension of the surface of porous ceramic materials

The fractal dimension D of the surface of porous ceramic materials has been determined. The dependence of D on the total porous space volume exhibits two bending points, which reflects a change in the character of porosity on the passage from isolated pores to connected pore clusters and to very large pores with smooth boundaries. This behavior reveals a correlation between the fractal dimension and some well-known features in the properties of porous solids.     

Fracture toughness and crack morphology in indentation fracture of brittle materials

The relationship between the indentation fracture toughness, K _c, and the fractal dimension of the crack, D, has been examined on the indentation-fractured specimens of SiC and AIN ceramics, a soda-lime glass and a WC-8%Co hard metal. A theoretical analysis of the crack morphology based on a fractal geometry model was then made to correlate the fractal dimension of the crack, D, with the fracture toughness, K _IC, in brittle materials. The fractal dimension of the indentation crack, D, was found to be in the range 1.024–1.145 in brittle materials in this study. The indentation fracture toughness, K _c, increased with increasing fractal dimension, D, of the crack in these materials. According to the present analysis, the fracture toughness, K _IC, can be expressed as the following function of the fractal dimension of the crack, D, such that $$In K_{IC} = {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}\{ In[2\Gamma E/(1 - \nu ^2 )] - (D - 1)In r_L \}$$ Where Γ is the work done in creating a unit crack surface, E is Young's modulus, v is Poisson's ratio, and r _L is r _min/r _max, the ratio of the lower limit, r _min, to the upper limit, r _max, of the scale length, r, between which the crack exhibits a fractal nature (r _min ?r?r _max). The experimental data (except for WC-8%Co hard metal) obtained in this study and by other investigators have been fitted to the above equation. The factors which affect the prediction of the value of K _IC from the above equation have been discussed.     

Study on wood fracture parallel to the grains based on fractal geometry

The fracture toughness (K _IC ) parallel to the grains of five kinds of wood was tested by compact tension specimen and the profile contour analysis method was employed to measure fractal dimensions D _s of their fracture surfaces. The results show that fracture toughness parallel to the grains of various woods is different because of their textural diversity and such differences are also shown on the morphology of fracture surfaces. Furthermore, the fractal dimension D _s and fracture toughness ${K_{IC}^{TL} }$ parallel to the grains have evident direct proportional relation, and this helps to reveal the inherent relationship between fracture toughness of wood and its microstructure.     

Fractality of tics as a quantitative assessment tool for Tourette syndrome

Tics manifest as brief, purposeless and unintentional movements or noises that, for many individuals, can be suppressed temporarily with effort. Previous work has hypothesized that the chaotic temporal nature of tics could possess an inherent fractality, that is, have neighbour-to-neighbour correlation at all levels of timescale. However, demonstrating this phenomenon has eluded researchers for more than two decades, primarily because of the challenges associated with estimating the scale-invariant, power law exponent—called the fractal dimension D_f—from fractional Brownian noise. Here, we confirm this hypothesis and establish the fractality of tics by examining two tic time series datasets collected 6–12 months apart in children with tics, using random walk models and directional statistics. We find that D_f is correlated with tic severity as measured by the YGTTS total tic score, and that D_f is a sensitive parameter in examining the effect of several tic suppression conditions on the tic time series. Our findings pave the way for using the fractal nature of tics as a robust quantitative tool for estimating tic severity and treatment effectiveness, as well as a possible marker for differentiating typical from functional tics.     

Modelling optical properties of soft tissue by fractal distribution of scatterers

Abstract

A knowledge of the local refractive index variations and size distribution of scatterers in biological tissue is required to understand the physical processes involved in light-tissue interaction. This paper describes a method for modelling the complicated soft tissue, based on the fractal approach, permitting numerical evaluation of the phase functions and four optical properties of tissue—scattering coefficient, reduced scattering coefficient, backscatter-ing coefficient, and anisotropy factor—by the use of the Mie scattering theory. A key assumption of the model is that refractive index variations caused by microscopic tissue elements can be treated as particles with size distribution according to the power law. The model parameters, such as refractive index, incident wavelength, and fractal dimension, that are likely to affect the predictions of optical properties are investigated. The results suggest that the fractal dimension used to describe how biological tissue can be approximated by particle distribution is highly dependent on how the continuous distribution is discretized. The optical properties of the tissue significantly depend on the refractive index of tissue, implying that the refractive index of the particles should be carefully chosen in the model in order accurately to predict the optical properties of the tissue concerned.     

The radial basis integral equation method for solving the Helmholtz equation

A meshless method for the solution of Helmholtz equation has been developed by using the radial basis integral equation method (RBIEM). The derivation of the integral equation used in the RBIEM is based on the fundamental solution of the Helmholtz equation, therefore domain integrals are not encountered in the method. The method exploits the advantage of placing the source points always in the centre of circular sub-domains in order to avoid singular or near-singular integrals. Three equations for two-dimensional (2D) or four for three-dimensional (3D) potential problems are required at each node. The first equation is the integral equation arising from the application of the Green’s identities and the remaining equations are the derivatives of the first equation with respect to space coordinates. Radial basis function (RBF) interpolation is applied in order to obtain the values of the field variable and partial derivatives at the boundary of the circular sub-domains, providing this way the boundary conditions for solution of the integral equations at the nodes (centres of circles). The accuracy and robustness of the method has been tested on some analytical solutions of the problem. Two different RBFs have been used, namely augmented thin plate spline (ATPS) in 2D and f(R)=⁴Rln(R) augmented by a second order polynomial. The latter has been found to produce more accurate results.     

Growth of fractal patterns in ZnO films doped with Fe

ZnO films both undoped and doped with Fe were deposited on Si substrates using rf-magnetron sputtering. The results showed that fractal features were clearly exhibited in the ZnO film doped with Fe. It is proposed that the fractal aggregates were the result of cluster diffusion-limited aggregation (CDLA) of magnetic particles on the surface of the film. The fractal dimension of a main branch (D=1.47) was smaller than that expected by the CDLA model (D=1.72). In this paper the growth mechanism of the observed fractal aggregates is discussed in terms of the magnetism of FeO, nanoparticle aggregation and surface tension changes.     

On estimating stress intensity factors and modulus of cohesion for fractal cracks

Existing solutions for the singular stress field in the vicinity of a fractal crack tip have been adapted for a somewhat modified problem. Since the integration along the fractal curve is prohibitive and does not lend itself to the presently available mathematical treatments, a simplified one has replaced the original problem. The latter involves a smooth crack embedded in a singular stress field, for which the order of singularity is adjusted to match exactly the one obtained from the analyses pertaining to the fractal crack. Of course, this is only an approximation, and we may only hope that it leads toward correct results, at least in a cursory sense. The advantage of such an approach becomes obvious when one inspects the final closed-form solutions for (a) the stress intensity factor in mode I fractal fracture, and (b) cohesion modulus, which results from the cohesive zone model and serves as a measure of the material resistance to crack propagation. As expected for the fractal geometry employed here, our results are strongly dependent on the fractal dimension D (or roughness exponent H).     

Synthesis, structure, and physical properties of carbon, iron, and chromium deposits possessing a fractal structure

A carbon deposit formed during graphite spraying in an electric arc, as well as iron and chromium particles obtained by electrochemical deposition under certain conditions, possess fractal structures. Data on some physical properties, the size of fractal aggregates, and the fractal dimension D of carbon, iron, and chromium deposits are presented. Relations between the fractal dimensions and physical properties of deposits are considered. A possible mechanism of the fractal structure formation is discussed.     

Some properties of solid fractal structures in carbon nanofibers

Experimental data on the self-organization of massive fractal granules with a total volume of up to 1.1 cm³ consisting of carbon nanofibers with transverse dimensions within 50–70 nm and a length of up to 1000 nm are presented. The fractal granules have a density of 1.3 g/cm³, an elastic modulus of 37.4 MPa, and a fractal dimension of D = 2.95. The resistivity of this material is about three orders of magnitude greater than the values typical of pyrolytic graphite. The surface layers in the fractal granule are characterized by a Seebeck coefficient of about S ～ 24 μV/K, whereas the bulk regions have S ～ 11 μV/K.     

Fractal modeling of the J-R curve and the influence of the rugged crack growth on the stable elastic-plastic fracture mechanics

In this paper, fractal geometry is used to modify the Griffith-Irwin-Orowan classical energy balance. Crack fractal geometry is introduced in the elastic-plastic fracture mechanics by means of the Eshelby-Rice J-integral and the influence of the ruggedness of the crack surface on the quasistatic crack growth is evaluated. It is shown that the rising of the J-R curve correlates to the topological ruggedness dimension of the crack surface. Results from fracture experiments are shown to be very well fitted with the proposed model, which is shown to be a unifying approach for fractal models currently used in fracture mechanics.     

Properties of Dense Assemblies of Magnetic Nanoparticles Promising for Application in Biomedicine

Chemically synthesized iron oxide nanoparticles and magnetosomes produced by magnetotactic bacteria are of great importance for application in biomedicine. In this paper, we discuss the complicated magnetic anisotropy of the nanoparticles, the influence of the magnetostatic interactions, and thermal fluctuations on the behavior of these assemblies. Numerical simulation for dilute assemblies of iron oxide nanoparticles with combined magnetic anisotropy show that the uniaxial shape anisotropy dominates even for small aspect ratios of the particle, L/D≥1.1–1.2. The quasistatic hysteresis loops are calculated for various clusters of bacterial magnetosomes with diameters D=40–60 nm to understand the influence of magnetostatic interactions. The specific absorption rate (SAR) is calculated for assemblies of magnetic nanoparticles dispersed in solid and liquid media. A new electrodynamic method of measurement is used to obtain the SAR of the assembly of bacterial magnetosomes with average diameter D=48 nm.     

Equations of anomalous absorption onto swelling porous media

Absorption is a very common process which takes place on various types of materials ranging from porous media to new nano-materials and biological tissues. The majority of studies reported on absorption to date are concentrated on “rigid” porous media, which contradict the properties of real porous media which undergo swelling and shrinking changes. Here we present new absorption equations derived from a fractional diffusion-wave equation (fDWE) for absorption onto swelling porous media in a material coordinate. We show that the cumulative anomalous absorption is I(t) = St^β/2 and the absorption rate , where S is the anomalous sorptivity and β the order of fractional derivative in fDWE. Using published data on cumulative absorption against time, the two adsorption parameters are determined: β = 1.2448 and S = 2.7775 cm²/h. The value of β = 1.2448 implies that absorption onto this swelling porous media belong to the category of super-diffusion, which is a phenomenon unknown to us before. In comparison, the traditional absorption equations do not have such features. When S is determined, the anomalous diffusivity, D_m, is calculated using its relation with S. We expect that the proposed new absorption equations will be valuable for explaining new phenomena and processes encountered in broader disciplines of science and engineering applications.     

An equilibrated method of fundamental solutions to choose the best source points for the Laplace equation

For the method of fundamental solutions (MFS), a trial solution is expressed as a linear combination of fundamental solutions. However, the accuracy of MFS is heavily dependent on the distribution of source points. Two distributions of source points are frequently adopted: one on a circle with a radius R, and another along an offset D to the boundary, where R and D are problem dependent constants. In the present paper, we propose a new method to choose the best source points, by using the MFS with multiple lengths R_k for the distribution of source points, which are solved from an uncoupled system of nonlinear algebraic equations. Based on the concept of equilibrated matrix, the multiple-length R_k is fully determined by the collocated points and a parameter R or D, such that the condition number of the multiple-length MFS (MLMFS) can be reduced smaller than that of the original MFS. This new technique significantly improves the accuracy of the numerical solution in several orders than the MFS with the distribution of source points using R or D. Some numerical tests for the Laplace equation confirm that the MLMFS has a good efficiency and accuracy, and the computational cost is rather cheap.     

The effect of contour angle on fractal dimension measurements for brittle materials

Slit-island analysis (SIA) has been successfully used to measure the fractal dimensional increment (D*) of fracture surfaces. The fracture toughness in brittle ceramics is related to the fractal dimension. However, the measurement technique may affect the determined D* values. The purpose of this study was to determine the contour angle at which a valid fractal dimension measurement could be obtained using the SIA method for baria silicate glass-ceramic and zinc selenide ceramic. Two specimens of each material were duplicated for each of the following contour angles: 0°, 3°, 5°, 7°, 22°, and 90°. After polishing to 1 m alumina slurry, the coastlines were photographed and arranged in a collage. The coastline was analyzed using the Richardson technique. Results showed that the SIA technique is sensitive to the contour angle since D* decreases with increasing contour angle for both materials.     

Fraunhofer Diffraction from Apertures Bounded by Regular Fractals

Abstract

Some properties of fields diffracted in the Fraunhofer region by apertures bounded by regular fractals are investigated. A recursion relation describing such apertures is introduced and the associated relation in the Fourier transform domain is described. For a triadic Koch aperture whose edge has the fractal dimension of D_s = 1·262, the recursion relation is numerically evaluated. Self-similar structures of intensity distributions in the Fraunhofer region are verified for the present objects. The relationship of the fractal dimension D _s of the fractal edge with the power-law decay of the Fraunhofer diffraction intensities is also verified.     

Modelling of rheological behaviour of semisolid metal slurries Part 4 – Effects of particle morphology

Abstract

In this part of the series, the fractal dimension D is introduced to quantify the morphology of equiaxed dendrites and the concept of equivalent solid fraction Ψ to connect solid fraction to the effective solid fraction. Generally, dendrite formed during diffusion controlled solidification has a fractal dimension of 2.5, while for a perfectly spherical particle D=3. The rheological model developed in Part 1 is then used to deduce the D data from experimental results of continuous cooling in the literature. It is shown that, within the range of experimental conditions concerned, D is predominantly affected by solid fraction (or semisolid temperature), while shear rate and cooling rate only have a limited effect on D. The rheological model developed in Part 1 is applied to study the rheological behaviour of semisolid metal (SSM) slurries with non-spherical particle morphology. It is found that under constant morphology assumption the shear thinning effect is much more pronounced for SSM slurries with solid particles of lower fractal dimension (i.e. a more dendritic morphology), and that the degree of thixotropy increases with decreasing fractal dimension of solid particles. In addition, it has been demonstrated that as a diffusion controlled process, particle spheroidisation takes many thousands of seconds, and is much slower than the deagglomeration process (a few seconds) and the agglomeration process (a few thousand seconds).     

Fractal and topological characterization of branching patterns on the fracture surface of cross-linked dimethacrylate resins

The branching patterns formed as a result of crack growth in dimethacrylate resins below their glass transition temperatures looked similar to fractal trees. The skeletons of the patterns were analysed numerically for their topological and geometrical properties. The number of branches, N _i , mean branch lengths, N _i , and branch angles of a particular order, defined according to the Strahler and inverted Weibel schemes, followed exponential scaling behaviour: N _i (R _b )^–i and L _i (R _l ) ⁱ . Using the relationship for the fractal dimension D=In R _B /In R _L , a value of D=1.4 was obtained for the fracture pattern. Fractal behaviour was also examined by the box-counting method which indicated a power-law dependence of the mass on the box size with fractal dimension exponent D=1.4 in the case of the fracture pattern. However, the mass-shell method for both the fracture pattern and the fractal trees gave an exponential increase of mass with distance from the origin, rather than the power-law behaviour expected for fractals. This was attributed to the fact that branches of different sizes were distributed in restricted regions of space closer to the periphery, rather than uniformly over the whole pattern.     

Correlation between crack tortuosity and fracture toughness in cementitious material

This paper proposes a new technique for the evaluation of fractal dimension (D) of fracture surface and a quantitative correlation between D and fracture toughness of cementitious materials. The experimental program has been performed on compact tension (CT) specimens (600 × 525 × 125 mm) with three different aggregate sizes (d _max=4.7 mm, 18.8 mm and 37.5 mm). The fractal geometry concept is utilized in the evaluation of fracture surface roughness. To avoid indirect or destructive experimental procedures that are prohibitively laborious and time consuming, a new non-destructive technique is presented. Results of the analysis indicate that the concept of fractal geometry provides a useful tool in the fracture surface characterization. The results also suggest that the fracture toughness can be correlated with the fractal dimension of fracture surface.