**Abstract**

The deformation of materials is a key topic for different industrial sectors. The correlation between specific thermomechanical processes, like extrusion, rolling or additive manufacturing, and the resultant material's microstructure, is particularly interesting. In these thermomechanical processes, the microstructure of the materials depends mainly on the applied stress, the magnitude of strain achieved in a given time period, and the temperature. In the case of metals and alloys, plastic deformation can be described microscopically based on the dynamics of a huge population of moving dislocations. Plasticity is characterized at the mesoscale by the avalanche-like collective behavior of dislocations, which is a typical case of self-organized systems. Dislocations are organized into cells named grains or subgrains that greatly influence the mechanical behavior of metals and alloys. The existence of these fractal structures of dislocations in metals is well established. However, it is very complex to conduct a fractal analysis of these microstructures. This is actually done by looking for a compromise between a detailed description of the complex fractal microstructure and the development of a practical procedure that avoids unaffordable extensive characterization in a different time and spatial scales. Several cases will be described considering different alloys and experimental conditions.

**Keywords:** fractal, metal, strain, dislocations, mesoscale

### **1. Introduction**

The concept of fractal refers to structures whose appearance is independent of the scale at which it is observed. In these geometrical objects, the same pattern appears regardless of whether we move away to have a more global vision of the observed structure or if we approach to expand a particular detail of it. Fractals are very common in many aspects of nature. They appear, for example, in the crystalline structure that develops in snowflakes or in the crystals that form frost during cold and wet sunrises. The rough, superficial structure of broccoli is another recurring example in living beings where a fractal nature is revealed. The structure of the veins through which the sage passes on the leaves of the trees is another very clear example of a fractal. The coastline can be described, in some cases, in the frame of a fractal object. Fractals appear not only in nature and are not restricted to physical objects, but they are also present in other fields, such in financial markets; the repeated and cyclic behavior of the stock market also falls in a fractal patter. Music is other field in which fractals have been reported, for example, when shorter passages are reflected in expanded form in longer passages. It is surprising that,

being fractals so common objects and present in so many fields, the concept of fractal was initially proposed as late as in 1975 [1]. This "self-similarity" of fractal geometry was proposed by the IBM mathematician Benoît Mandelbrot (the term fractal derives from the Latin word *fractus*).

arrangement of dislocations in metals has been proposed to be associated with a fractal structure. Examples are found in various usual metal forming processes such

*Fractal Analysis of Strain-Induced Microstructures in Metals*

*DOI: http://dx.doi.org/10.5772/intechopen.91456*

a large number of dislocations in the generation of fractal structures [10]. The sample size is one of the features that determine the fractal dimension of these dislocation structures [11]. In that study, carried out in iron micropillars, it has been observed that the fractal dimension of the dislocation structure changes approximately 40%, from 1.1 to 1.5, when the sample diameter increases from 300 to 1500 nm. This reveals the strong size effect on fractal substructures of dislocations

**2. Origin of fractal structures in metals: stochastic vs. thermodynamic**

Materials, in particular metals, can be considered complex systems from the mechanical point of view, considering that they contain a large number of elements and dislocations that interact with each other. A complex system has properties that arise from the interaction between its elements and the system itself with the environment. In particular, one of these properties is the emergency. The emergency can be defined as the generation of spontaneous patterns resulting from the self-organization of the elements of the system. Classic examples of an emergency in nature are snowflakes. These patterns have different geometries with a high degree of symmetry indicating the ability to organize the system. Surprisingly, a similar behavior has been found in the case of gallium-based liquid metals. By applying an electric potential to this liquid metal, snowflake-shaped fractals have been found [12]. These structures have encouraged the introduction of the term fractalized metals. It has been proposed that this concept defines a new state in which metals can be found. In this state, all positions in the metal lattice are

Metal microstructures have a great amount of defects. Dislocations, linear defects, are the most important ones from the point of view of plastic deformation and mechanical behavior. An individual, isolated, dislocation has an associated elastic stress field. For an edge dislocation, is described, in Cartesian coordinates, by

<sup>2</sup>*π*ð Þ <sup>1</sup> � <sup>ν</sup> *<sup>x</sup>*<sup>2</sup> <sup>þ</sup> *<sup>y</sup>*<sup>2</sup> ð Þ<sup>2</sup> ; *<sup>σ</sup>xy* <sup>¼</sup> *<sup>σ</sup>yx*

<sup>2</sup>*π*ð Þ <sup>1</sup> � <sup>ν</sup> *<sup>x</sup>*<sup>2</sup> <sup>þ</sup> *<sup>y</sup>*<sup>2</sup> ð Þ<sup>2</sup> ; *<sup>σ</sup>zz* <sup>¼</sup> <sup>ν</sup> *<sup>σ</sup>xx* <sup>þ</sup> *<sup>σ</sup>yy* ; *<sup>σ</sup>xz* <sup>¼</sup> *<sup>σ</sup>zx* <sup>¼</sup> *<sup>σ</sup>zy* <sup>¼</sup> *<sup>σ</sup>yz* <sup>¼</sup> 0 (1)

where *G* is the shear modulus, *b* is the Burgers vector, ν is Poisson's ratio, and *x*, *y*, and *z* are the coordinates, perpendicular in-plane, perpendicular out-of-plane, and parallel directions, with respect to the dislocation line, respectively, in the

The specific stress field depends on the character of the dislocation, i.e., screw or edge. The stress fields of a huge amount of dislocations, in combination with the external applied stress, play a relevant role in the formation of fractal deformation patterns in metals. Through these stress fields, interactions occur with other dislocations or with second phases of the material, such as precipitates. The interaction among typically 1010–10<sup>14</sup> m�<sup>2</sup> dislocations generates fractal patterns in the

coordinated granting properties such as chiral transport [13].

a tensor where the components, *σij*, are given by Eq. (1) [14].

<sup>2</sup>*π*ð Þ <sup>1</sup> � <sup>ν</sup> *<sup>x</sup>*<sup>2</sup> <sup>þ</sup> *<sup>y</sup>*<sup>2</sup> ð Þ<sup>2</sup> ; *<sup>σ</sup>yy* ¼ � *Gby x*<sup>2</sup> <sup>þ</sup> *<sup>y</sup>*<sup>2</sup> ð Þ

*<sup>σ</sup>xx* ¼ � *Gby* <sup>3</sup>*x*<sup>2</sup> <sup>þ</sup> *<sup>y</sup>*<sup>2</sup> ð Þ

<sup>¼</sup> *Gbx x*<sup>2</sup> � *<sup>y</sup>*<sup>2</sup> ð Þ

reference system of the dislocation.

**105**

It is important to understand the role of the collective motion and interaction of

as rolling [9].

in metals.

**approach**

The widespread appearance of the concept of fractal in many areas is also observed in different microstructural features of materials. In fact, the appearance of fractal structures in metals is widely accepted [2]. Solid state metals have an orderly crystalline structure that provides their physical properties. The strongest metal bond is formed when the metallic atoms are ordered as close as possible, usually in cubic or hexagonal lattices. These lattices are described by the unit elemental cell characterized by the lattice parameter, dimension measured in nm and their symmetry typically represented by cubic or hexagonal unit cells. Most of the metals we find in our daily lives are polycrystals. This means that not all the unit elemental cells of the material have got the same orientation with respect each other. Actually, there are domains of atoms/unit cells where their orientation is the same. These domains, which are present in millions in a metallic component, are called grains. These grains are separated from each other by grain boundaries. All this microstructural complexity is a fertile breeding ground for fractal structure appearance. One example is the presence of fractals in the solidification/crystallization structure generated in the manufacturing process of polycrystalline metals during cooling [3]. Another example of fractal structures in metals is provided by the distribution of reinforcement in discontinuously reinforced composite materials. The physical properties of these composite materials are governed by the properties of the clusters that are formed when contacts are established among the reinforcing particles. The percolation of the system greatly modifies its properties. The formation of these clusters depends on the fractional content of the composite material [4]. Another classic example of fractal structures in metals is the fracture surface [5]. These fracture surfaces are generated from the interaction of macroscopic defects, mainly cracks due to the application of mechanical stresses that generate a progressive deformation of the material until its breakage. These surfaces are formed by facets of different sizes and heights that can be accurately described by a fractal. The knowledge of the fractal nature of fracture surfaces has allowed the design of fractal surfaces to achieve certain properties such as a better air flow avoiding turbulence [6].

The fractal nature of the fracture surfaces is an indication that the deformation process generates fractal structures in the deformed microstructure, which ultimately give rise to the fractal fracture surface. In the case of metals, the deformation process begins with an elastic deformation that occurs in small intervals typically smaller well below 0.2% strain. The stress value achieved at this point is known as the yield stress, and it has been agreed that it reflects the beginning of plastic deformation of metals in the macroscale (homogeneous deformation). From this strain/stress value, the deformation is mainly caused by the movement of dislocations. The main defects in metals are vacancies and dislocations. These defects correspond to the lack of a semi-plane of atoms in its crystalline structure, respectively. When the temperature is high and the applied stress is low, this movement can be controlled by the motion of vacancies (diffusion). Dislocations are found not only inside the polycrystalline grains but also at the grain boundaries. A characteristic of dislocations, the fundamental element in metal deformation, is that their collective arrangement has a fractal nature. Sevillano et al. [7] were the first to recognize the possibility of fractal geometry of the cell structure based on the measurement of the fractal dimension of the microstructure of cold worked Cu. These dislocations are organized during the deformation process and form cell structures that have a fractal nature [8]. Moreover, the tangled

*Fractal Analysis of Strain-Induced Microstructures in Metals DOI: http://dx.doi.org/10.5772/intechopen.91456*

being fractals so common objects and present in so many fields, the concept of fractal was initially proposed as late as in 1975 [1]. This "self-similarity" of fractal geometry was proposed by the IBM mathematician Benoît Mandelbrot (the term

The widespread appearance of the concept of fractal in many areas is also observed in different microstructural features of materials. In fact, the appearance of fractal structures in metals is widely accepted [2]. Solid state metals have an orderly crystalline structure that provides their physical properties. The strongest metal bond is formed when the metallic atoms are ordered as close as possible, usually in cubic or hexagonal lattices. These lattices are described by the unit elemental cell characterized by the lattice parameter, dimension measured in nm and their symmetry typically represented by cubic or hexagonal unit cells. Most of the metals we find in our daily lives are polycrystals. This means that not all the unit elemental cells of the material have got the same orientation with respect each other. Actually, there are domains of atoms/unit cells where their orientation is the same. These domains, which are present in millions in a metallic component, are called grains. These grains are separated from each other by grain boundaries. All this microstructural complexity is a fertile breeding ground for fractal structure appearance. One example is the presence of fractals in the solidification/crystallization structure generated in the manufacturing process of polycrystalline metals during cooling [3]. Another example of fractal structures in metals is provided by the distribution of reinforcement in discontinuously reinforced composite materials. The physical properties of these composite materials are governed by the properties of the clusters that are formed when contacts are established among the reinforcing particles. The percolation of the system greatly modifies its properties. The formation of these clusters depends on the fractional content of the composite material [4]. Another classic example of fractal structures in metals is the fracture surface [5]. These fracture surfaces are generated from the interaction of macroscopic defects, mainly cracks due to the application of mechanical stresses that generate a progressive deformation of the material until its breakage. These surfaces are formed by facets of different sizes and heights that can be accurately described by a fractal. The knowledge of the fractal nature of fracture surfaces has allowed the design of fractal surfaces to achieve certain properties such as a better air flow

The fractal nature of the fracture surfaces is an indication that the deformation process generates fractal structures in the deformed microstructure, which ultimately give rise to the fractal fracture surface. In the case of metals, the deformation process begins with an elastic deformation that occurs in small intervals typically smaller well below 0.2% strain. The stress value achieved at this point is known as the yield stress, and it has been agreed that it reflects the beginning of plastic deformation of metals in the macroscale (homogeneous deformation). From this strain/stress value, the deformation is mainly caused by the movement

of dislocations. The main defects in metals are vacancies and dislocations. These defects correspond to the lack of a semi-plane of atoms in its crystalline structure, respectively. When the temperature is high and the applied stress is low, this movement can be controlled by the motion of vacancies (diffusion). Dislocations are found not only inside the polycrystalline grains but also at the grain boundaries. A characteristic of dislocations, the fundamental element in metal deformation, is that their collective arrangement has a fractal nature. Sevillano et al. [7] were the first to recognize the possibility of fractal geometry of the cell structure based on the measurement of the fractal dimension of the microstructure of cold worked Cu. These dislocations are organized during the deformation process and

form cell structures that have a fractal nature [8]. Moreover, the tangled

fractal derives from the Latin word *fractus*).

*Fractal Analysis - Selected Examples*

avoiding turbulence [6].

**104**

arrangement of dislocations in metals has been proposed to be associated with a fractal structure. Examples are found in various usual metal forming processes such as rolling [9].

It is important to understand the role of the collective motion and interaction of a large number of dislocations in the generation of fractal structures [10]. The sample size is one of the features that determine the fractal dimension of these dislocation structures [11]. In that study, carried out in iron micropillars, it has been observed that the fractal dimension of the dislocation structure changes approximately 40%, from 1.1 to 1.5, when the sample diameter increases from 300 to 1500 nm. This reveals the strong size effect on fractal substructures of dislocations in metals.
