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This work addresses the role of rigid inclusions in determining the elastic modulus of particle-filled colloidal networks by modifying an established fractal scaling model. The approach acknowledges the heterogeneous nature of stress distribution at length scales beyond the colloidal aggregates, while maintaining structural information at the level of individual clusters. This was achieved by introducing a scaling factor to account for system heterogeneity which contains intrinsic information about the network’s capacity to form load-bearing links. Rigid fillers bound to the network induce stress concentration, but additionally serve as junction zones which introduce additional load-bearing pathways. This gives rise to the observed increase in the modulus with filler volume fraction. The proposed relationship between the load-bearing network connectivity and scaling behavior may have additional implications on the fractal dimension determined by rheological methods. Further, this model accommodates an experimentally observed correlation between the scaling behavior of the modulus associated with the addition of fillers and that arising from increasing structurant concentration. The modified fractal model thus provides an alternative view of how fillers contribute to the small- and large-deformation mechanical behavior of filled colloidal gels in a manner consistent with experimental observations.

Investigation of rock progressive damage under static confinement and strain rates facilitates the generation mechanism of natural fault damage zones. A triaxial Hopkinson bar apparatus is used to perform dynamic triaxial compression tests to examine the damage and degradation process of rocks subjected to multiple impacts. Dynamic mechanical properties are determined under a static triaxial pre-stress of (30, 20, 10) MPa and multiple dynamic loadings, with the repetitive impact velocity of 27 m/s and strain rates from 50 to 150/s. The acoustic characteristics are identified by ultrasonic measurement to qualify the damage values. The micro-crack parameters, including crack area and volumes are detected using synchrotron X-ray micro-computed tomography (μCT) to characterize the progressive damage. In addition, the microcrack orientation, density and fractal dimension are analysed from thin section. Experimental results show that dynamic stress-strain curves can be divided to elastic, nonlinear deformation and unloading phases. Dynamic peak stress, Young’s modulus and ultrasonic wave velocity decrease with increasing impact times. The high frequency of ultrasonic wave is filtered by the induced microcracks. The progressive damage and evolution of fracture networks are associated highly with microcrack initiation, propagation, branching and coalescence. Shear bands are commonly generated in granite, and tensile cracks are dominant in marble, while sandstone is mainly failed by compaction and deformation band. The absorbed energy of rock increases nonlinearly with increasing crack surface and volume. Besides, microcracks propagate primarily along the maximum principal stress; the density and fractal dimension exhibit an anisotropic distribution controlled by true triaxial confinement and dynamic impacts.

The basic structural factors enabling the design of high-modulus and high-strength nanocomposites (polymer/carbon-nanotube) with mechanical characteristics analogous to the corresponding parameters of steel are studied in the framework of fractal analysis and percolation theory. The main factor is the formation in the polymer matrix of carbon-nanotube tows consisting of many of them with a structural fractal dimension close to Euclidean (close to three). Such tows have an elastic modulus close to the nominal value of a separate nanotube. The proposed procedure allows the limiting mechanical characteristics of nanocomposites to be determined.

We consider physical foundations for the realization of high-modulus and high-strength polymer/carbon nanotube nanocomposites with mechanical characteristics comparable with those for steel. We have determined two main factors that make it possible to obtain such nanocomposites, i.e., the structure of the nanofiller in the polymer matrix and a quite high concentration of the nanofiller. The fractal dimension of such a structure must be close to that of the surrounding Euclidian space (i.e., 3). It is shown that additional stretching of the nanocomposite gives two positive effects: the reduction of waviness of carbon nanotubes and the increase of the elastic modulus of the polymer matrix due to alignment of its macromolecules.

Cold rollings of A1008 steel specimens are carried out in this study to analyze the influences of roll’s surface pattern and specimen’s reduction ratio on surface morphologies, optical properties, and mechanical properties of rolled samples. A 3D fractal formula, along with dedicated algorithms developed, is applied to solve the periodic (characteristics) lengths and fractal dimensions for roll and rolled surfaces along and perpendicular to the rolling directions. The correlation for every fractal parameter between roll and the rolled specimen is developed to be a function of roll’s surface pattern and reduction ratio first. The light tracking simulations were then carried out for rolled samples only. Rockwell hardness and tensile tests were conducted to evaluate the specimen’s hardness and Young’s modulus. A roll having a higher fractal dimension, or a longer periodic length, can result in a higher fractal dimension and a longer periodic length of the rolled specimen, respectively. A larger reduction ratio also results in the rolled surface with higher fractal dimensions and longer periodic lengths. Increasing the fractal dimensions of a rolled specimen can elevate the maximum illuminance and, conversely, lower the minimum illuminance, while an increase in periodic lengths can increase the minimum illuminance and decrease the maximum illuminance. A decrease in the fractal dimensions and/or increase in the periodic lengths of rolled surfaces in the two directions can raise the illuminance uniformity. For the specimens prepared with the same roll pattern, an increase in reduction ratio can bring an increment in illuminance uniformity and hardness of rolled samples. This study provides an efficient way to determine the roll pattern and reduction ratio that satisfies the specific demands in optical properties.

The bonding surface structure generated by the repair of concrete structures has been paid more attention as a weak point. The effects of old concrete age, interface roughness, and freeze-thawing (F-T) attack on adhesive interface are comprehensively investigated. In this study, six kinds of interface roughness and five different old concrete age are designed. The interfacial bonding property is mainly evaluated by splitting tensile strength (fts). Fractal analysis was used to characterize the interface roughness using laser scanning data. In general, the fts increased with the increasing value of interface fractal dimension. The relationship between fts and fractal dimension value was further analyzed, considering the old concrete ages and the F-T cycles. The results show that the effect of roughness on the bonding property of new-to-old concrete is more significant than the age of old concrete, and the influence of the F-T cycles on the bonding surface is mainly reflected in the initial stage of the F-T deterioration process. The relative dynamic elasticity modulus decreased obviously under F-T cycles, especially for the specimens with low interface roughness. In combination with the results of two non-destructive methods (ultrasonic non-destructive test and relative dynamic elastic modulus test), the larger roughness and the smaller age of old concrete can improve the bond performance.

A large number of post mortem human subjects was used to investigate the relation between the micro-structure of rib cortical bone and the mechanical properties using Fractal Dimension. Uniaxial tensile tests were performed on coupons of rib cortical bone. Tensile strength, yield stress, Young's Modulus, maximum strain, and work to fracture were determined for each coupon. Fractal dimension was computed using CT images and Digital Image Correlation procedures. A highly significant effect of fractal dimension in the mechanical properties was found. In addition, the variation in mechanical properties was found to be adequately represented by Generalized Extreme Value type distributions.

Just as it is regarding ordinary cement-based materials, the pore structure and microstructure of alkali-activated materials are disordered. It is essential to predict the macroscopic properties by studying the pore and microstructure fractal characteristics of materials. In this paper, the effects of slag content and alkali activator modulus on compressive strength, porosity, and microstructure of alkali-activated coal gangue-slag (AACGS) mortar were studied. Further, with the help of mercury intrusion porosimetry (MIP) data and the MATLAB programming, the pore and SEM photos fractal dimensions of AACGS mortar specimens were obtained, respectively, and the relationship between the microscopic fractal dimensions and the macroscopic strength and the structural characteristics of pores was established. The results show that the pore fractal dimension has a good linear relationship with the compressive strength and pore characteristic parameters (porosity, total pore area, and average pore diameter, etc.). With the increase of slag content, the SEM photos fractal dimension of AACGS mortar specimens increases, and the fractal dimension and compressive strength also show a significant positive linear relationship. The two fractal characterization methods can be used in the alkali-activated material system and have important guiding significance for predicting the macroscopic strength and pore characteristic parameters of the material.

Microcontact affects the fabrication and assembly of MEMS/NEMS system, the contact between thin film microcavity and microcantilever beam, and the motion of microstructure. In this work, in the microcontact of three-dimensional elastic-plastic Weierstrass-Mandelbrot (W-M) fractal surfaces, influence of fractal dimension was studied based on a comprehensive contact model. With increasing fractal dimension, maximum microcontact force in the plastic deformation zone shows parabolic change; comparably, the intermediate force in elastic zone parabolically varies. For both the forces, the minimum values are obtained when the fractal dimension is 2.5. Besides, in the plastic deformation zone, the real contact areas increase with the fractal dimension. Experiments were completed to compare with the numerical analysis. The results show that the simulated contact force curve is in line with the experimental load curve when Young’s modulus E and hardness H are equal to the actual measured values. Nevertheless, it will greatly deviate from the experimental load curve when E and H differ from the measured values.

On the basis of the M-B fractal theory, a geometric model and contact mechanical model of the tooth surface contact mode were analyzed. According to the energy dissipation mechanism of tangential contact damping, a tangential contact damping model of tooth surface considering variable friction under dry friction condition was established by introducing the correction coefficient of tooth surface. First, obtained the load and tangential damping of a micro-convex body, and then the contact load and tangential contact damping of the tooth surface are obtained by integrating on the tooth surface. It was shown that the tangential contact damping of tooth surface is affected by various parameters including roughness amplitude, fractal dimension, curvature radius of tooth surface, normal contact load and tangential normal load ratio. The simulation results show that the tangential contact damping gradually increases when the normal contact load, friction coefficient, curvature radius of driving wheel and roughness amplitude increase. The tangential contact damping on the tooth surface increased with the increase in the tangential normal load ratio. When the tangential normal load ratio is 0.09, the tangential contact damping of the tooth surface reached a maximum, and then decreased with the increase of the tangential normal load ratio. The change in the tooth surface tangential contact damping with the fractal dimension is similar to the change in the tangential normal load ratio, which also increased and then decreased. When the fractal dimension is approximately 1.64, the tangential contact damping of the tooth surface reached the maximum. The tangential contact damping of tooth surface increases with the increase of modulus, transmission ratio and the number of teeth of the driving gear, and the number of teeth and modulus have a great influence on it, while the influence of transmission ratio is relatively weak.

In this paper the mechanical properties of silica aerogels are studied by simulations with the usage of the Movable Cellular Automata (MCA) method. In the MCA approach a sample is considered as a set of interacting cellular automata. The procedure to prepare porous samples is based on random spheres method. The resulting structures have densities in the range from 111 to 2200 kg/m3 and porosity up to 95%. The structures of the samples are studied by obtaining the fractal dimension using the FD3 program. The Young's modulus is examined in the simulation of a compression test. The elastic modulus relate to density by a power law characterized by exponent 3.83 ± 0.18 for samples with densities from 111 to 568 kg/m3 and 3.17 ± 0.11 for samples with densities from 350 to 2200 kg/m3. A comparison to previously published data from experiments and comparable computer simulations is included. The results prove that the Movable Cellular Automata method is suitable to investigate the mechanical properties of silica aerogels.

Polymer-modified bitumen (PmBs) has been characterised by fluorescence microscopy, conventional performance tests and by the rheological measurements, before and after the Rolling Thin Film Oven Test (RTFOT), and by multi-fractal analysis. The addition of 2–7 wt-% of linear styrene-butadiene-styrene block copolymer (SBS-L) or radial block copolymer (SBS- R) increased viscosity, softening point and complex modulus of PmBs. Hardening of PmBs occurred and the elastic response decreased after RTFOT aging. SBS-L PmBs are more susceptible to RTFOT aging. Digital fluorescence micrographs of sections of PmBs qualitatively exhibit different morphological structures which can be quantitatively probed by multi-fractal analysis: differences are reflected in the f (α) spectrum. The PmBs which appear less multi-fractal have the best rheological properties. In another approach, the focal fractal dimensions’ distributions for the SBS-modified bitumen have peak around 2.35 (which corresponds to the bitumen-rich phase) and around 2.7 (which corresponds to the noncontinuous SBS-rich phase).

Mechanical properties of metallic foams are highly dependent on the characteristics of porosity. In the case of foams manufactured using space holder particles (SHPs), parameters such as percentage porosity, pore size and cell wall thickness are directly contingent on the SHP used. In this work, different fractal distributions of SHPs were introduced in order to modify the resulting porosity. An Al–Si–Cu alloy was used as raw material for producing metallic foams by infiltration, while NaCl particles of 2 and 4 mm were used as SHPs, modifying the small-to-large particle ratio and fractal dimensions. Cylindrical foams of 10 cm in height and 5 cm in diameter were obtained by infiltrating the Al alloy into the SHPs. Results showed that the use of fractal distribution allowed to slightly increase the percentage porosity, whereas the most important effect was the decrease in cell wall thickness when fractal dimension increased. Mechanical properties were directly affected by these modifications, decreasing when fractality was induced. Finite element analysis models of the foams were obtained in a combination with discrete element method, in order to simulate their compressive elastic behaviors. Young’s modulus estimations were in excellent agreement with experimental results, validating the effect of fractal inclusion and the importance of the correct model selection.

Proppants for hydraulic fracturing usually refer to quartz sand, ceramic proppants and their modified products as well. Importantly, the fracture conductivity varies with the performance and properties of different proppants. However, there are still some uncertainties between the material properties and fracture conductivity. To this end, a numerical model was established based on the distribution of proppant particle size, material properties, fractal dimension of pore structure, elastic deformation, proppant embedment according to fractal theory and principle of elastic-plastic mechanics as well, which is used to predict fracture conductivity under multi-factor. The results show that fracture conductivity is directly related to material properties between proppant and reservoir rock. The Young’s modulus of rock has a significant effect on the conductivity of the propped fracture. And, the influence of Poisson’s ratio is weak. The fracture conductivity increases with the increase of pore fractal dimension.

The pore structure in char particles changes greatly during combustion, which significantly affects the char combustion characteristics. This article describes a model for the evolution of the char pore structure. A fractal pore model is used to generate the porous char particles to simulate the char combustion for various porosities, , specific surface areas, , fractal dimensions, , and particle diameters, . Then, expressions are given to associate these pore structure parameters with the char conversion, , and the operating conditions, using a modulus, , indicating the pore diffusion resistance. As increases, increases, and decrease, and increases to a maximum first and then decreases. In some cases, only decreases with increasing when the reaction rates are too high or the conditions are such that the O2 cannot diffuse into the particle. The results compare well with the pore structure parameters of three Chinese chars combusted in a drop tube furnace at different conversions measured by a merc...

The effect of varying pore structures on the kinetics of SO2–CaO reactions is not fully understood in the previous studies. Combining fractal pore model, gas molecular movement model and two-stage reaction model, a new desulfurization model is established in this paper. Fractal pore model is used to simulate CaO particle and gas molecular movement model is used to simulate gas diffusion in pores. Fractal dimension is used to characterize complexity of pore structure instead of tortuosity factor. It is found that the reaction is significantly affected by pore structures. A modulus ϕ is introduced to characterize the relationship between varying pore structures and apparent reaction parameters. And this relationship is verified by thermo-gravimetric analysis (TGA) data. Comparing to the previous models, the effect of varying pore structure on the kinetics of the reaction is described more accurately by the desulfurization model.

Linear and nonlinear viscoelasticity of gelatin solutions was investigated by rheology. The dynamic mechanical properties during the sol-gel transition of gelatin followed the time-cure superposition. The fractal dimension df of the critical gel was estimated as 1.76, which indicated a loose network. A high sol fraction ws = 0.61 was evaluated from the plateau modulus by semi-empirical models. Strain-stiffening behavior was observed under large amplitude oscillatory shear (LAOS) for the gelatin gel. The strain and frequency dependence of the minimum strain modulus GM, energy dissipation Ed, and nonlinear viscoelastic parameter NE was illustrated in Pipkin diagrams and explained by the strain induced helix formation reported previously by others. The BST model described the strain-stiffening behavior of gelatin gel quite well, whereas the Gent and worm-like chain network models overestimated the strain-stiffening at large strains.

a b s t r a c t Fractal dimension (FD) can be used to characterize microstructure of porous media, particularly bone tissue. The porous microstructure of cortical bone is observable in micro-CT (CT) images. Estimations of fractal dimensions of CT images of coupons of human cortical bone are obtained. The same sam- ples were tested on a tensile test machine and Young's modulus (YM) and Failure stress were obtained. When both types of measures were compared, a clear correlation was found (R = −81%, P < 0.01). Young's modulus of each sample and the FD of its CT images are correlated. From the assumption that cortical bone is approximately a fractal set, a non-linear constitutive relation involving FD is obtained for YM. Experimental results show good agreement with this constitutive relation. Additional parameters in the non-linear relation between YM and FD have been estimated from experimental results and related to physical parameters.

Fractal pore models, gas molecular movement models, and reaction models are combined to simulate char oxidation in char pores. The numerical results show the relationship between the pore structure and the apparent char reaction parameters. A modulus number φ is introduced to characterize the diffusion properties in fractal porous media with a relationship given for the apparent reaction parameter in terms of φ. The results show that the char combustion process is strongly impacted by the pore structure, and that the fractal dimension is an important parameter that should be considered in char combustion. Comparison with experimental data shows that this model is more accurate than the classical intrinsic combustion model.

Deformation memory effect (DME) is one of the rock memory effects. One important application of the DME is to determine the in situ stress state. Compared to the traditional in situ stress measurements, the methods based on the DME are commercial and permit large number of measurements. Application of DME needs enough reliable identification methods. However, the existing methods sometimes are indistinct and the amount is insufficient. combined with three traditional methods including tangential modulus method, deformation rate analysis (DRA), acoustic emission method, two new potential methods were explored in the stress region above crack initiation threshold. One is based on the fractal dimension, called FD method. Another one is to take advantage of the lateral strain in the DRA method and the FD method, instead of using the axial strain. Based on the contact bond model in PFC 2D , numerical model for granite sample was developed and cyclic uniaxial compressions were performed on it. Both the existing methods and new methods were used to detect the DME. The results demonstrate that the FD method is effective and reliable, result by DRA method with lateral strain is better than that with the axial strain, the tangential modulus method is not so distinct as other methods.

According to frame concrete theory, Concrete was composed of mortar matrix, coarse aggregate, and transition zone between mortar matrix and coarse aggregate. Proper particle grading of fine aggregate can not only save cement, but also increase comprehensive properties of mortar matrix. Uniform design method was used to study the relationships between the grading of fine aggregate and workability, and the relationship with mechanic properties of mortar matrix and concrete. Based on fractal theory, the relationship between fractal dimension and modulus of fineness, and the relationship between fractal dimension and n value in Fuller equation were researched. It concluded that grading of fine aggregate affects the workability and mechanic properties of mortar matrix and concrete, significantly. There was a good linear relation between fractal dimension and n value in Fuller Equation, and fratatal dimension decreased with modulus of fineness increasing.

The electrical resistivity, Young’s modulus, and microhardness of a Cu-10 at % Ni alloy preliminarily deformed by rolling at a reduction from 0 to 89% are studied after isochronous annealing in the temperature range 100–800°C. It is found that a decrease in the electrical resistivity of the alloy upon annealing at temperatures up to 500°C is accompanied by an increase in the microhardness and Young’s modulus, which, in turn, is caused by short-range decomposition of the alloy with the formation of nickel atom clusters. A possible fractal nature of these clusters is discussed. The number of particles in these clusters can vary from 2 to 10, and the fractal dimension changes from 1.10 to 2.89, respectively.

A method of synthesis of fractal nanostructure micrographs most similar to the initial electron micrographs is considered. The method is based on the modified modulus of the Fourier transform of the examined micrographs; moreover, information on the transformation phase is preserved. The nanostructure micrographs so obtained can be filtered out from false structures engendered by some methods of electron-microscopic study.

Model systems having different microstructures and rheological properties were obtained by controlled crystallization from a mixture of high-melting and low-melting lipids. Based on analysis of confocal scanning light microscopic images, the microstructural characteristics of the systems were quantified by use of different approaches including microstructure density, Euler characteristic, nearest-neighbor analysis, fractal dimension of microstructure interface, and fractal dimension by the particle-counting method (PCM). The solid-fat content (SFC) of semisolid lipid samples was measured by nuclear magnetic resonance (NMR) spectroscopy and rheological properties were analyzed by compressive penetration tests with a texture analyzer. As expected, SFC had a major impact on rheological properties, but lipid crystalline microstructure also had significant effects. Correlation analysis showed that rheological properties were highly correlated with the various quantitative microstructural parameters, with the exception of the fractal dimension by the PCM. Empirical models adequately correlated rheological properties with SFC and microstructure density. Compression modulus increased by a factor of about ten as SFC increased from 0.28 to 0.51. However, for systems with the same SFC, compression modulus was dependent on microstructure. At low SFC compression modulus increased by about a factor of seven over the range of microstructures formed, whereas at higher SFC compression modulus only increased by a factor of about two.

The amount of particle breakage of Tacheng rockfill material (TRM) in the process of shearing was quantitatively measured by a relative breakage index. The fractal dimension of TRM at the final state of the tests was found to be linearly related to the void ratio and logarithm of the confining pressure. The influences of the relative breakage index on the friction angle, modulus, and deformation were investigated based on a series of large-scale triaxial compression tests on TRM. An increase in the relative breakage index leads to an increase in the volumetric strain and to decreases in the peak state friction angle, critical state friction angle, deviatoric strain, and critical state void ratio. The test results were used to propose simple formulations pertaining to the relative breakage index for the strength and deformation indexes (e.g., peak state friction angle, critical state friction angle, initial elastic modulus, secant modulus at 50% of the peak strength, volumetric strain, deviatoric s...

The change in elastomer tensile moduli, as formulated in the Gaussian statistical theory of rubber elasticity, with deformation, is considered both experimentally and theoretically. Gum elastomers of different structures and corresponding materials filled with carbon black, as reinforcing filler, are investigated experimentally. For all materials considered, the same scaling pattern with negative and low slope for small deformations, and positive and higher slope for large deformations is obtained, indicating two distinct mechanisms of elastic response. Most pronounced is the similarity of small deformation responses for filled materials. Considering the modulus as an elastic energy density gradient dependent on structure changes with deformation, and interpreting the changes for small deformations in terms of conformational energy change, the fractal dimension of a new type is formulated. It describes the decrease in elastomer network connectivity with deformations, which is discussed in terms of conformon dynamics. Possibilities of application of Faynman's path integral method and statistical method of random walk to the lattice are considered for the conformon, as well.

The rheological behavior of adsorption layers of β-casein formed at the air/water interface was studied by bubble tensiometry. It was observed that in some instances, the dilational modulus, e, has only an elastic component while in other cases it has both an elastic and a viscous component. The conditions where the elastic modulus is significant have been defined. In these conditions, the measured dilational modulus is practically equal to a physical quantity measured in equilibrium conditions. Until a surface pressure, π, of about 5 mN/m, it is proportional to the surface pressure. Referring to a multiblock polymer model, the proportionality coefficient y (e = yπ) is directly related to the fractal dimension of the polymer forming a pancake at the interface. An increase of temperature from 10 to 60 °C results in a decrease of y from 6.6 to 3.3, indicating that in these conditions the two-dimensional polymer changes from a partly collapsed structure to that of a two-dimensional self-avoiding walk. These ...

In this paper, we generalize Barenblatt's cohesive fracture theory for fractal cracks. We discuss the difficulties of generalizing the concept of traction on a fractal surface. Borodich's modification of Griffith's theory for fractal cracks is reviewed. Irwin's driving force is generalized for fractal cracks and a fractal driving force (Gf) is defined. It is shown that to generalize Barenblatt's theory for fractal cracks it is necessary to introduce a new quantity, D-fractal cohesive pseudo-stress. This new quantity is cohesive force per unit of a fractal measure. Fractal modulus of cohesion is seen to be a function of both the material and the fractal dimension of the crack. Equivalence of fractal Barenblatt's and Griffith's theories is discussed. It is seen that the order of stress singularity at the tip of a fractal crack cannot be obtained using modified Barenblatt's theory because this theory is a local theory and assumes the order of stress singularity a priori.

The purpose of this study was to compare the structural parameters of fabric and fractal dimension as predictors of the Young's modulus of equine cancellous bone. Eight 15 mm cubes of cancellous bone were obtained from three equine third metacarpal bones. Young's modulus was determined for the three orthogonal directions. The fabric and fractal dimension were calculated for each of the six exposed faces of each cube. Fractal dimension plus porosity provided a higher explanatory power for Young's modulus (R2 = 78.7%. P < 0.0001) than fabric plus porosity (R2 = 69.2%, P < 0.0001). Fractal dimension was also significantly correlated with fabric (R2 = 53.8%, P < 0.0001). Although this novel method for combining fractal dimension data into a pseudo-directionally dependent predictor of Young's modulus requires further validation over a greater range of porosities and differing cancellous bone tissues, its potential has been demonstrated.

Foamed del has begun to play an important role in permeability modification applications because of the reduced chemical requirements. Foamed gels create impermeable barriers in porous media; however, once a critical pressure differential is exceeded, the permeability increases with increasing pressure. A two-dimensional network model was developed to estimate foamed gel barrier performance in terms of the maximum pressure a barrier can withstand and the evolution of the foamed gel barrier's permeability. The formation of conductive pathways and the accopanying permeability increase were estimated from a model of the pressure-induced deformation and repture of individual lenses. The evolution of conductive pathways changed from invasion percolation (high elastic modulus, rigid gel) to a lens rupture chain reaction initiated by teh rupture of a single lens (low elstic modulus gel) as the elastic modulus of the gel was decreased. The apparent fractal dimension of the first conductive channel ranged from 1.89 to 1.06 for high and low wlastic modulus gels, respectively. This dependency of breakthrough and breakdown is unique and produces a large range of breakdown behavior for any degree of microscopic heterogeneity.

The value of the critical exponent for modulus (t) was estimated by using a model based on the percolation theory, and the t value was experimentally determined for poly(vinyl alcohol) (PVA) gels. The model proposed in this study has shown that there is a critical strain (γc), which distinguishes two regimes in the elastic response of the gel network near the gelation point. The theory has also shown that γc is infinitesimally small for a realistic gelation in the 3D space. The exponent t for γ⪢γc scales as (d+dmin−dB)ν≈2.27 with fractal dimensions, dmin and dB, and the exponent ν for correlation length for gels near the gelation point. The value of t for PVA gels has been found to be 2.31±0.04 and is close to the one predicted by the model proposed.

The influence of the physical properties of a cross-linked polyester resin on its electrical treeing behaviour in the point-plane geometry is investigated. The physical properties of the resin were varied by post-curing, and these were characterized by thermal analysis and mechanical modulus measurements. Electrical trees were grown under constant geometry and stress conditions and their fractal dimension and initiation and propagation statistics determined. These were then correlated with the changes in the polymer network characteristics. Comparisons with recent stochastic modelling are also discussed briefly.

An improved understanding of the processing and mechanical behaviour of fibre-reinforced composites would extend the potential applications of these materials. For unidirectional composites, the type of packing or the degree of randomness affects the transverse modulus and localised absence of fibres is related to longitudinal compression failure. Woven reinforcements permit more effective manufacture than individual aligned fibres. Compression strengths of woven composites are reduced when fibres are clustered. Clustering of fibres is predicted to increase the resin flow in the laminate and hence expedite the processing of these materials. Commercial fabrics employ this concept using flow-enhancing bound tows. The net effect of clustering fibres is to enhance processability whilst reducing the mechanical properties. This process-property dilemma was the focus of our research in the last decade. The effects reported above were qualitative correlations. To improve the design tools for reinforcement fabrics we have sought to quantify the changes in the micro-/meso-structure of woven reinforcement fabrics. The use of automated image analysis is essential for the quantification of subtle changes in fabric architecture. The classification of structured populations can be achieved by a variety of parameters. This paper considers Voronoi tessellation and fractal dimensions for the quantification of the microstructures of woven fibre-reinforced composites and reviews the process-property-structure relationships for commercial and experimental materials studied experimentally in an attempt to resolve the processing versus properties dilemma.

Crumpled graphite thin film balls were fabricated with Panasonic Pyrolytic Graphite Sheets (PGS). The fractal dimension, mechanical properties, and electrical conductivity of the crumpled PGS balls have been investigated. The universal local fractal dimension of the PGS balls is found to be 2.58, which is consistent with that of paper balls. The crumpled PGS balls show good mechanical property with Young's Modulus of 16–17 N, which is about the same as that of paper balls and elastoplastic paper balls, but with much smaller sizes, thinner film thicknesses, and less weight. In addition, the crumpled PGS balls show good conductivity, slightly higher than that of the PGS film before crumpling.

Recent work on the rheology of gels formed by the irreversible aggregation of spherical colloidal particles is reviewed briefly with an emphasis on scaling. The properties considered include the gel modulus and strength, non-linear viscoelasticity, shear-flow and the effect of flow on structure.

Critical behavior in gelation and vulcanization of polymers is discussed in the context of polymeric fractals. Dynamic mechanical measurements show that in the critical region a power law in the complex viscosity and in the modulus appears. Viscosity and modulus are calculated around the critical point and as functions of the extent of reaction. The power law is interpreted in terms of the fractal dimension of the cluster, and it is shown that this agrees with experiment if one takes swelling phenomena into account.

Cab-O-Sil powder is a cluster aggregate of silica spheres formed from silica smoke. The fractal dimension of clusters of 14 nm diameter spheres is determined to be [Formula: see text], in good agreement with that calculated for the cluster-aggregate model, [Formula: see text]. Measurements of Young's modulus (Y) of packed 7 nm Cab-O-Sil powder, with an occupied volume fraction in the range 0.035 c = 0.017 ± 0.002 and an elasticity exponent τ = 2.9 ± 0.2. The measured exponent is larger than the calculated scalar Born (balls and springs) result, τ = 2.0, but smaller than the more realistic bond-bending result, τ > 3.55. The measured result is attributed to bond bending within a complex percolation structure.