Fractal Inclusion Research Articles

Effect of fractal distribution of the porosity on mechanical properties of Al foams manufactured by infiltration

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.

Thermal properties of composite two-layer systems with a fractal inclusion structure

In this work, we study the thermal transport properties of platelike composite two-layer samples made of polyester resin and magnetite inclusions. By means of photoacoustic spectroscopy and thermal relaxation, their effective thermal diffusivity and conductivity were experimentally measured. The composite layers were prepared under the action of a static magnetic field, resulting in anisotropic (fractal) inclusion structures with the formation of chain-like magnetite aggregates parallel to the faces of the layers. In one kind of the bilayers, a composite layer was formed on top of a resin layer while their relative thickness was varied. These samples can be described by known models. In contrast, bilayers with the same concentration of inclusions and the same thickness on both sides, where only the angle between their inclusion structures was systematically varied, show a nontrivial behaviour of their thermal conductivity as a function of this angle. Through a multifractal and lacunarity analysis, we explain the observed thermal response in terms of the complexity of the interface between the layers.

Interaction of chitosan with benzoic acid in solution and films

Chitosan films are prepared from solutions containing benzoic acid. Depending on preparation conditions, films of two types can be prepared: homogeneous optically transparent films and biphasic films containing fractal inclusions in a transparent matrix. The presence of ionic interaction between chitosan and benzoic acid in films is confirmed with the use of IR spectroscopy. The algorithm of the analysis of fractal dimensionality based on halftone images is suggested.

Phononic band gap crystals with periodic fractal inclusions: Theoretical study using numerical analysis

A numerical study is conducted using finite difference time-domain analysis to determine the impact of periodic fractal-shaped inclusions on the frequency response of two-dimensional phononic or acoustic band gap crystals. Both solid host with solid inclusions and solid host with fluid inclusions were studied with increasing fractal order iteration or increasing fractal dimensionality for different types of fractals generated from square inclusions in a square lattice formation. Solid-fluid systems showed marked changes in frequency response including increasing multiplicity of bands compared to regular square inclusion shapes. Fractal inclusions in the solid host/fluid inclusion systems studied exhibited the ability to provide band gaps for much longer wavelengths than those of the solid host/solid inclusion systems studied. This behavior was attributed to the possibility of resonant localization effects that a fractal fluid inclusion can exhibit compared to fractal solid inclusions.

Numerical calculations of the intrinsic electrostatic resonances of artificial dielectric heterostructures

In order to study the intrinsic electrostatic resonances (ERs) of artificial dielectric heterostructures, we develop an efficient effective-medium-based method for modeling the effective permittivity, with careful attention paid to several key factors controlling ERs. Our method relies on finite element modeling and is applicable to inclusions with complex boundaries, e.g., fractal inclusion. A series of isolated and square arrays of several types of negative-permittivity media is considered. The inclusion shapes investigated can be considered as cross sections of infinite three-dimensional objects, where the properties and characteristics are invariant along the perpendicular cross-sectional plane. The continuum model used in this work is accurate only if the homogeneous representation of the composite structure makes sense, i.e., quasistatic approximation. It is found, among the conclusions of the article, that the effective permittivity of the composite (lossless) structures versus surface fraction curves presents a sharp peak, which occurs precisely at ER. For lossy inclusions, the primary signature of the ER is seen in the peak in the imaginary part of the complex permittivity or as an inflexion in the curve of the real part of the complex permittivity. The focus in this effort is on the analysis of intrinsic ER as a function of the shape and permittivity of the inclusion. The variations in the effective permittivity related to the iteration number show the following hierarchy for Sierpinski’s square and triangle: the higher the iteration number of the inclusion the smaller value of ϕ2 corresponding to the ER. In the vicinity of the ER peak, field enhancement is observed, which consists of enormous changes in the local electric field. Differences between the ER characteristics for aperiodic and periodic orders through the introduction of localized voids in the structure are also noteworthy. In addition, our approach performs well for fractal-shaped inclusions, e.g., Siepinski square, for which we show that the ERs satisfy a similarity transformation. These calculations can aid in the discovery of new materials with optimized magnetoelectric structures whose ER may be manipulated by electromagnetic fields.

Dielectric response of perforated two-dimensional lossy heterostructures: A finite-element approach

Finite-element simulations of the effective complex permittivity of perforated two-dimensional (2D) lossy heterostructures are reported. The method is computationally inexpensive and is suited for simulations where the tacit assumptions are the following: (1) the composite behaves like a homogeneous medium with an effective (relative) permittivity ε=ε′−jε″ and (2) the porous medium is characterized by a perforated 2D object having arbitrary shape, e.g., split rings, honeycomb lattice, and Sierpinski carpet. These shape functionals have many applications to the scattering of wave and are also important for describing effective properties of particle dispersions. Our calculations provide insights into a variety of tuning parameters influencing ε including the surface fraction and perimeter of inclusion, the permittivity contrast between the inclusion and the matrix, and the shape of the holes. For a 2D composite structure containing a deterministic fractal inclusion we explicitly demonstrate that the ε′ and ε″ changes with reduced perimeter can be modeled according to the same similarity transformation, at least for the first four iterations of the fractal pattern. We quantify the effect of increasing the internal porosity on ε′ and ε″ for different types of perforated structures and show that composites containing split rings can achieve very small ε′ and ε″ in a large range of porosity. We find also that such geometries are auspicious for local field enhancement. The origin of these enhancements lies in the breakdown of the dipolar approximation which is often used to evaluate the effective permittivity of composite materials. These results can provide the experimenter with a method for assessing permittivity measurements and suggest that the successful integration of voided material in microwave electronic devices depends on the morphology of the embedded porosity.