Elasticity, fractal structures

We now examine the main ideas of how we define the conductivity and elasticity of fractal structures. 

In order to calculate the elastic properties of fractal structures according to the iterative procedure the Hashin-Strikman double-sided estimation of the elastic properties can be used [131], namely,... 

Before we discuss the viscoelastic properties of nonuniform fractal structures we shall give some basic definitions from the theory of elasticity. 

Flocculation occurs when the net force between the particles is attractive. At low volume fractions, aggregation results in clusters, or floes, which have a fractal structure (7). For most systems, the properties of the aggregating suspension changes drastically at a certain critical particle concentration, 0g, which corresponds to the formation of a space-filling particle network. In dilute suspensions, at 0 < 0g, suspensions have no yield stress and the discrete clusters will settle more or less independently. Above 0g, the suspension can sustain a stress before yielding the elasticity may be significant, and the rate of settling is very slow. 

Keywords Nanocomposite high-elasticity matrix filler aggregation strain localization fractal structure. 

The first models, describing elastic behavior of fractal structures, were used, as a rule, for simulation within the fimneworks of percolation theory [1-5], Nonhomogeneous statistical mixture of solid and liquid then only displays solid body properties (e g., not equal to zero shear modulus G), when solid component forms percolation cluster, like at gelation in pol5uner solutions. If liquid component there is replaced by vacuum, then bulk modulus. B will also be equal to zero below percolation threshold [1]. Such model gives the following relationship for elastic constants [1,3] ... 

Hence, the stated above results show that the classical theory of entropic high-elasticity can be used for the description of stress-draw ratio curves for rubbers with weak strain hardening, but it is incorrect in case of nanocomposites with elastomeric matrix. The correct description of deformation behavior of the latter gives the high-elasticity fractal model that is due to fractal nature of filled rubbers structure [13]. 

As a rule, at present crosslinked polymer networks are characterised within the frameworks of entropic rubber high-elasticity concepts [2, 3]. However, in recent years works indicating a more complex structure of crosslinked rubbers have appeared. Flory [4] demonstrated the existence of dynamic local order in rubbers. Balankin [5] showed principal inaccuracy of the entropic high-elasticity theory and proposed a high-elasticity fractal theory of polymers. These observations suppose that more complete characterisation of these materials is necessary for the correct description of the structure of rubbers and their behaviour at deformation. In paper [6] this was carried out by the combined use of a number of theoretical physical concepts, namely the rubber high-elasticity entropic theory, the cluster model of the amorphous state structure of polymers [7, 8] and fractal analysis [9]. 

The first models describing the elastic behaviour of fractal structures used, as a rule, simulation within the frameworks of percolation theory [21-25]. Anon-homogeneous statistical mixture of solid and liquid displays solid properties (for instance, shear modulus G not equal to zero) only, when the solid component forms a percolation cluster at gelation in polymer solutions. If the liquid component is replaced by a vacuum then the bulk modulus K. will also be equal to zero below the percolation threshold [21]. This model gives the following relationship for elastic constants [21] ... 

Ferri, F., Greco, M., Arcovito, G., De Spirito, M., and Rocco, M. (2002). Structure of fibrin gels studied by elastic light scattering techniques Dependence of fractal dimension, gel crossover length, fiber diameter, and fiber density on monomer concentration. Phys. Rev. E. 66(1 Pt 1), 011913. 

The behaviour and magnitude of the storage and loss moduli and yield stress as a function of applied stress or oscillatory frequency and concentration can be modelled mathematically and leads to conclusions about the structure of the material.3 For supramolecular gels, for example, their structure is not simple and may be described as cellular solids, fractal/colloidal systems or soft glassy materials. In order to be considered as gels (which are solid-like) the elastic modulus (O ) should be invariant with frequency up to a particular yield point, and should exceed G" by at least an order of magnitude (Figure 14.2). 

Zolotuhin I.V. et al. (1998) Structure, internal friction and modulus of elasticity of fractal carbon deposit. Fiz. Tberd. Tela 3, 584-586. (In Russian). 

Accordingly, we expect a power law behavior G,0 (O/Op)3 5 of the small strain elastic modulus for 0>0. Thereby, the exponent (3+df [j)/(3—df)w3.5 reflects the characteristic structure of the fractal heterogeneity of the filler network, i.e., the CCA-clusters. The strong dependency of G 0 on the solid fraction Op of primary aggregates reflects the effect of structure on the storage modulus. 

The established concepts predict some features of the Payne effect, that are independent of the specific types of filler. These features are in good agreement with experimental studies. For example, the Kraus-exponent m of the G drop with increasing deformation is entirely determined by the structure of the cluster network [58, 59]. Another example is the scaling relation at Eq. (70) predicting a specific power law behavior of the elastic modulus as a function of the filler volume fraction. The exponent reflects the characteristic structure of the fractal heterogeneity of the CCA-cluster network. 

Marangoni s group has since advanced the fractal theory applying Shih et al. s weak link regime with Vreeker s rheological findings to develop a fractal theory for fat crystal networks. Fat crystal networks are considered as cross-linked fractal clusters formed by aggregating fat crystals. Self-similarity is assumed to exist within the clusters, from the primary fat crystals to the clusters. If the force-constant of the links between micro structures was expressed as k/, then the macroscopic elastic constant K (in one dimension) of the network could be modeled as ... 

We have also studied the elastic properties of a nonuniform medium with chaotic structure in which one phase has a negative shear modulus. The analysis may be made using the fractal hierarchical structure model. 

Fractal dimension, D is considered as an effective number that characterises the irregular electrode surface. The term has been related to physical quantities such as mass distribution, density of vibrational stages, conductivity and elasticity. If we consider a 2-D fractal picture in its self-similar multi-steps, one can draw various spheres of known radii at various points of its structure and may thus count the number of particles, N inside the sphere by microscope, following relation will then hold good ... 

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