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

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 coordinated granting properties such as chiral transport [13].

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

$$\begin{split} \sigma\_{\text{xx}} &= -\frac{\text{Gby}(3\mathbf{x}^2 + \mathbf{y}^2)}{2\pi(1-\nu)(\mathbf{x}^2 + \mathbf{y}^2)^2}; \sigma\_{\text{\mathcal{V}}} = -\frac{\text{Gby}(\mathbf{x}^2 + \mathbf{y}^2)}{2\pi(1-\nu)(\mathbf{x}^2 + \mathbf{y}^2)^2}; \sigma\_{\text{xy}} = \sigma\_{\text{\mathcal{V}xx}} \\ &= \frac{\text{Gbx}(\mathbf{x}^2 - \mathbf{y}^2)}{2\pi(1-\nu)(\mathbf{x}^2 + \mathbf{y}^2)^2}; \sigma\_{\text{xx}} = \nu \left(\sigma\_{\text{xx}} + \sigma\_{\text{\mathcal{V}}}\right); \sigma\_{\text{xx}} = \sigma\_{\text{xx}} = \sigma\_{\text{\mathcal{V}x}} = \sigma\_{\text{\mathcal{V}x}} = \mathbf{0} \end{split} \tag{1}$$

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 reference system of the dislocation.

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

mesoscale. The formation of these fractals follows the same fundamental rules as any other physical system which develops a fractal structure. Here, it is particularized to the specific characteristics of metals. The formation of fractals in physical systems can be explained from a stochastic or a thermodynamic point of view. The concept that represents the susceptibility of uniform systems to spontaneously develop patterns in uniform nonlinear systems is known as Turing instability [15]. These patterns are found in reaction–diffusion systems describing, for example, the transformation of substances by chemical reactions in which the substances are transformed among each other and spread out over by diffusion. In the case of metals, a reaction– diffusion model of coupled nonlinear equations has been used to describe the formation of forest (immobile) and gliding (mobile) dislocation densities in the presence of cyclic loading [16]. This description is done assuming that a metal undergoing both an internal and external stress is far from equilibrium. The second law of thermodynamics, states that any physical system can spontaneously increases its entropy, i.e., from order to disorder. The apparent conflict between this postulate and the generation of ordered fractal patterns is dismissed by considering that the accompanying decrease in entropy (i.e., ordering) is compensated by a corresponding increase somewhere else, so that the net entropy always increases. However, very recent theories [17] postulate that the apparent conflict between the second law of thermodynamics and ordered systems evolution is due to the consideration that the Universe is near equilibrium. This new theory postulates that there is an energy/matter flow comprising the whole Universe that evolves from disorder to order via selforganization. This process must be described by the empirical laws of nonequilibrium thermodynamics [18]. In this situation, a stress gradient in relatively simple non-equilibrium systems, as deformed metals, causes a flux of energy/matter in the system. As a consequence, it emerges a countervailing gradient. These conjugated processes result in spatio-temporal macroscopic order that spontaneously emerges provided that system is driven far away enough from equilibrium [19, 20]. Some authors have recently accepted that the contribution of the macroscopic (or mesoscopic) order, described by the entropy, must be considered for a rigorous description of the mechanical behavior of metals at high temperature [21].

measure of complexity for systems that do not present integer dimensions [25]. However, there are different fractals that have the same fractal dimension. Therefore, other parameters have been proposed to describe some characteristics of fractals. One of the most important characteristics of fractal structures of dislocations generated during metal deformation is their connectivity. Dislocations are usually arranged in grain and subgrain boundaries. These boundaries maintain their integrity, while the deformation increases due to the stress fields of dislocations that lock them in low-energy configurations. The connectivity of these structures can be described in this context by the distance to next dislocation neighbors. In particular, lacunarity is one of the most used parameters to characterize the connectivity among different parts of a fractal structure. Lacunarity is a measure of spatial heterogeneity, and it is used to differentiate images with similar fractal dimension but different appearances [26]. The local connected fractal dimension represents the connectivity of the different parts of the system by a color map that indicates

All parameters that describe a fractal structure of dislocations can be calculated from images obtained from the microstructure. In the deformation of metals, the most interesting images are related with the observation of dislocations that can be homogeneously distributed or rearranged to form subgrains or grain boundaries. These dislocation structures have been described in a very detailed manner from decades thanks to transmission electron microscopy (TEM). However, there are some aspects that must be taken into account when analyzing the fractal parameters of dislocation microstructures. The first one is the quality of the images. Normally, metallographical images have a gray range. In the case of a TEM image, moreover, there is a great contrast evolution depending on the interaction between the electrons and defects (dislocations, stacking faults, etc.), the relative orientation of the crystal, and the mode in which the images are obtained (bright field vs. dark field). **Figure 1** shows a TEM bright field image of the microstructure of a magnesium alloy which was deformed at intermediate temperatures. The presence of dislocations is clearly observed. The fractal dimension is typically measured from this kind

The "box counting" is a method for analyzing complex patterns by splitting an image into smaller and smaller pieces, typically "box"-shaped. The fractal dimension is calculated considering the number of boxes containing part of the microstructure vs. the number of empty boxes. The ratio of full vs. empty boxes is considered for at least 4/5 box sizes for the analysis of the fractal dimension. Computer-based box counting algorithms have been applied to patterns in one-, two-, and three-dimensional spaces [27, 28]. The technique is usually implemented in specialized software to analyze patterns extracted from digital images. This software also has application in the determination of related parameters such as

The TEM photos must be treated to obtain a binary image. However, in this process, TEM images cannot be completely clean of artifacts such as shadows or dots. The presence of these artifacts introduces an overestimation of the fractal

**Figure 2** shows several idealized examples presenting this situation. **Figure 2(a)** shows an idealized grain before deformation. Therefore, no dislocations or subgrain boundaries appear. After deformation, we can calculate the fractal dimension, DF, of, as an exercise, one or two dislocations within this grain (**Figure 2(b)** and **(c)**, respectively). For these two ideal examples, the fractal dimension calculated with

the proximity of the neighbor pixels in a given image.

*Fractal Analysis of Strain-Induced Microstructures in Metals*

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

of images using the box counting method.

lacunarity and multifractal analysis [29].

dimension.

**107**

**3.1 Image analysis**

Stochastics models have been also used to justify the appearance of fractal patterns developed during strain of metals. The necessity of a stochastic treatment, specific for fractal structures, for the characterization of mesoscopic dislocation network distributions in metals was established at the very end of the twentieth century [22]. At temperatures below 0.5 Tmelt (with Tmelt, the melting temperature), it is considered that there is no time-dependent plasticity (creep). In this range of temperatures, plasticity is explained by the avalanche-like collective behavior of dislocations [23]. The self-organized structures of dislocations generated by this phenomenon are developed at the grain size scale. This can be described as the mesoscale in metals. Natural nondeterministic fractals are self-similar in a statistical sense over a wide range of scales [1]. The localization of the plastic deformation of metals and alloys into slip bands has been investigated for a long time [24]. One example of dislocation patterns developed under uniaxial tensile tests appears with the Lüders band formation in some steels. The appearance of these patterns under uniaxial stress conditions reveals the great importance of dislocations interaction in their formation.

#### **3. Experimental characterization of fractal structures**

There are different parameters that can be used to describe a fractal. The most common one is the fractal dimension. Fractal dimensionality is considered a

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

measure of complexity for systems that do not present integer dimensions [25]. However, there are different fractals that have the same fractal dimension. Therefore, other parameters have been proposed to describe some characteristics of fractals. One of the most important characteristics of fractal structures of dislocations generated during metal deformation is their connectivity. Dislocations are usually arranged in grain and subgrain boundaries. These boundaries maintain their integrity, while the deformation increases due to the stress fields of dislocations that lock them in low-energy configurations. The connectivity of these structures can be described in this context by the distance to next dislocation neighbors. In particular, lacunarity is one of the most used parameters to characterize the connectivity among different parts of a fractal structure. Lacunarity is a measure of spatial heterogeneity, and it is used to differentiate images with similar fractal dimension but different appearances [26]. The local connected fractal dimension represents the connectivity of the different parts of the system by a color map that indicates the proximity of the neighbor pixels in a given image.

#### **3.1 Image analysis**

mesoscale. The formation of these fractals follows the same fundamental rules as any other physical system which develops a fractal structure. Here, it is particularized to the specific characteristics of metals. The formation of fractals in physical systems can be explained from a stochastic or a thermodynamic point of view. The concept that represents the susceptibility of uniform systems to spontaneously develop patterns in uniform nonlinear systems is known as Turing instability [15]. These patterns are found in reaction–diffusion systems describing, for example, the transformation of substances by chemical reactions in which the substances are transformed among each other and spread out over by diffusion. In the case of metals, a reaction– diffusion model of coupled nonlinear equations has been used to describe the formation of forest (immobile) and gliding (mobile) dislocation densities in the presence of cyclic loading [16]. This description is done assuming that a metal undergoing both an internal and external stress is far from equilibrium. The second law of thermodynamics, states that any physical system can spontaneously increases its entropy, i.e., from order to disorder. The apparent conflict between this postulate and the generation of ordered fractal patterns is dismissed by considering that the accompanying decrease in entropy (i.e., ordering) is compensated by a corresponding increase somewhere else, so that the net entropy always increases. However, very recent theories [17] postulate that the apparent conflict between the second law of thermodynamics and ordered systems evolution is due to the consideration that the Universe is near equilibrium. This new theory postulates that there is an energy/matter flow comprising the whole Universe that evolves from disorder to order via selforganization. This process must be described by the empirical laws of nonequilibrium thermodynamics [18]. In this situation, a stress gradient in relatively simple non-equilibrium systems, as deformed metals, causes a flux of energy/matter in the system. As a consequence, it emerges a countervailing gradient. These conjugated processes result in spatio-temporal macroscopic order that spontaneously emerges provided that system is driven far away enough from equilibrium [19, 20]. Some authors have recently accepted that the contribution of the macroscopic (or mesoscopic) order, described by the entropy, must be considered for a rigorous description of the mechanical behavior of metals at high temperature [21].

*Fractal Analysis - Selected Examples*

Stochastics models have been also used to justify the appearance of fractal patterns developed during strain of metals. The necessity of a stochastic treatment, specific for fractal structures, for the characterization of mesoscopic dislocation network distributions in metals was established at the very end of the twentieth century [22]. At temperatures below 0.5 Tmelt (with Tmelt, the melting temperature), it is considered that there is no time-dependent plasticity (creep). In this range of temperatures, plasticity is explained by the avalanche-like collective behavior of dislocations [23]. The self-organized structures of dislocations generated by this phenomenon are developed at the grain size scale. This can be described as the mesoscale in metals. Natural nondeterministic fractals are self-similar in a statistical sense over a wide range of scales [1]. The localization of the plastic deformation of metals and alloys into slip bands has been investigated for a long time [24]. One example of dislocation patterns developed under uniaxial tensile tests appears with the Lüders band formation in some steels. The appearance of these patterns under uniaxial stress conditions reveals the great importance of

dislocations interaction in their formation.

**106**

**3. Experimental characterization of fractal structures**

There are different parameters that can be used to describe a fractal. The most

common one is the fractal dimension. Fractal dimensionality is considered a

All parameters that describe a fractal structure of dislocations can be calculated from images obtained from the microstructure. In the deformation of metals, the most interesting images are related with the observation of dislocations that can be homogeneously distributed or rearranged to form subgrains or grain boundaries. These dislocation structures have been described in a very detailed manner from decades thanks to transmission electron microscopy (TEM). However, there are some aspects that must be taken into account when analyzing the fractal parameters of dislocation microstructures. The first one is the quality of the images. Normally, metallographical images have a gray range. In the case of a TEM image, moreover, there is a great contrast evolution depending on the interaction between the electrons and defects (dislocations, stacking faults, etc.), the relative orientation of the crystal, and the mode in which the images are obtained (bright field vs. dark field).

**Figure 1** shows a TEM bright field image of the microstructure of a magnesium alloy which was deformed at intermediate temperatures. The presence of dislocations is clearly observed. The fractal dimension is typically measured from this kind of images using the box counting method.

The "box counting" is a method for analyzing complex patterns by splitting an image into smaller and smaller pieces, typically "box"-shaped. The fractal dimension is calculated considering the number of boxes containing part of the microstructure vs. the number of empty boxes. The ratio of full vs. empty boxes is considered for at least 4/5 box sizes for the analysis of the fractal dimension. Computer-based box counting algorithms have been applied to patterns in one-, two-, and three-dimensional spaces [27, 28]. The technique is usually implemented in specialized software to analyze patterns extracted from digital images. This software also has application in the determination of related parameters such as lacunarity and multifractal analysis [29].

The TEM photos must be treated to obtain a binary image. However, in this process, TEM images cannot be completely clean of artifacts such as shadows or dots. The presence of these artifacts introduces an overestimation of the fractal dimension.

**Figure 2** shows several idealized examples presenting this situation. **Figure 2(a)** shows an idealized grain before deformation. Therefore, no dislocations or subgrain boundaries appear. After deformation, we can calculate the fractal dimension, DF, of, as an exercise, one or two dislocations within this grain (**Figure 2(b)** and **(c)**, respectively). For these two ideal examples, the fractal dimension calculated with

can be developed. This increase in dislocation density results in an additional increase in the fractal dimension (**Figure 2(d)** and **(e)**). As it is shown in **Figure 2 (b)**–**(e)**, the fractal dimension of dislocation substructures is related to the area covered by dislocations/substructure. Then, the fractal dimension of the structure shown in **Figure 2(e)** formed by thick lines is 10% greater than the structure shown in **Figure 2(b)**. However, there are some elements, mainly dots, which greatly influence the fractal dimension calculated by the box counting technique. As an example, **Figure 2(f)** shows the same structure as **Figure 2(c)**, as far as it is assumed that during the binarization process, the dusty dots are part of the image. These dusty dots are considered as an artifact. As a result of the consideration of the

*Fractal Analysis of Strain-Induced Microstructures in Metals*

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

dusty points in the image, the fractal dimension increases around 25%.

invisible when g = [0002] (a) is excited and visible with g = [1011] (b).

There are other aspects that must be taken into account when the fractal dimension of a dislocation structure in a deformed metal is calculated using TEM images. The first one is related to the fact that TEM images of the dislocations can be taken under several 3D orientations of the crystal and/or using different diffraction vectors. Dislocations can be visible only when the scalar product g b (where g is the diffraction vector of the crystal that is excited, i.e., Bragg's conditions are attained) is different from zero. **Figure 3** shows an example of a magnesium alloy deformed at room temperature. The Burgers vector in magnesium is [1120]. Dislocations are

The second one is that TEM images can be obtained in different modes known as bright and dark fields. **Figure 4** shows the image of a deformed sample of a magnesium alloy obtained in the same area, using the same crystal orientation

B = [1120] and the same diffraction vector g = [1011]. The difference between both images is that image in **Figure 4(a)** was obtained in dark field mode (image is generated with a diffracted spot) (weak beam), while image in **Figure 4(b)** was obtained in bright field mode (image is generated with the transmitted spot).

Within the magnesium grain, different defects have been created during plastic

deformation: dislocations, twins, and subgrains. Twins are clearly observed in bright field mode (**Figure 4b**), and dislocations and subgrains are clearly observed in dark field mode. Therefore, the fractal dimension of the complete deformed substructure changes depending on the TEM mode selected. In general, a combination of images must be used for a complete description of the microstructure. However, there are mechanical tests conducted under specific experimental

*TEM image of a magnesium alloy deformed at 3% at room temperature and a strain rate of 10<sup>3</sup> s*

*1*

*. Zone axis*

**Figure 3.**

**109**

*[1120] (a) [0002] and (b) g = [1101].*

#### **Figure 1.**

*Bright field image of a magnesium alloy deformed around 2% in tensile mode at a temperature of 200°C and a strain rate of 10*�*<sup>3</sup> s* �*1 . Zone axis [1120] g = [1100].*

#### **Figure 2.**

*(a) Idealized microstructure of a completely dislocation-free grain before plastic deformation. (b–d) Subgrain formation during the deformation process. (e) Subgrain coarsening during deformation process. (f) Microstructure shown in (c) with an artifact.*

the plugin FracLac [30] used in FIJI software is very similar (0.96 and 0.99, respectively). Since dislocations are line defects, these values are very reasonable. As the macroscopic strain increases, new dislocations are generated, or, even, subgrains

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

can be developed. This increase in dislocation density results in an additional increase in the fractal dimension (**Figure 2(d)** and **(e)**). As it is shown in **Figure 2 (b)**–**(e)**, the fractal dimension of dislocation substructures is related to the area covered by dislocations/substructure. Then, the fractal dimension of the structure shown in **Figure 2(e)** formed by thick lines is 10% greater than the structure shown in **Figure 2(b)**. However, there are some elements, mainly dots, which greatly influence the fractal dimension calculated by the box counting technique. As an example, **Figure 2(f)** shows the same structure as **Figure 2(c)**, as far as it is assumed that during the binarization process, the dusty dots are part of the image. These dusty dots are considered as an artifact. As a result of the consideration of the dusty points in the image, the fractal dimension increases around 25%.

There are other aspects that must be taken into account when the fractal dimension of a dislocation structure in a deformed metal is calculated using TEM images. The first one is related to the fact that TEM images of the dislocations can be taken under several 3D orientations of the crystal and/or using different diffraction vectors. Dislocations can be visible only when the scalar product g b (where g is the diffraction vector of the crystal that is excited, i.e., Bragg's conditions are attained) is different from zero. **Figure 3** shows an example of a magnesium alloy deformed at room temperature. The Burgers vector in magnesium is [1120]. Dislocations are invisible when g = [0002] (a) is excited and visible with g = [1011] (b).

The second one is that TEM images can be obtained in different modes known as bright and dark fields. **Figure 4** shows the image of a deformed sample of a magnesium alloy obtained in the same area, using the same crystal orientation B = [1120] and the same diffraction vector g = [1011]. The difference between both images is that image in **Figure 4(a)** was obtained in dark field mode (image is generated with a diffracted spot) (weak beam), while image in **Figure 4(b)** was obtained in bright field mode (image is generated with the transmitted spot).

Within the magnesium grain, different defects have been created during plastic deformation: dislocations, twins, and subgrains. Twins are clearly observed in bright field mode (**Figure 4b**), and dislocations and subgrains are clearly observed in dark field mode. Therefore, the fractal dimension of the complete deformed substructure changes depending on the TEM mode selected. In general, a combination of images must be used for a complete description of the microstructure. However, there are mechanical tests conducted under specific experimental

#### **Figure 3.**

*TEM image of a magnesium alloy deformed at 3% at room temperature and a strain rate of 10<sup>3</sup> s 1 . Zone axis [1120] (a) [0002] and (b) g = [1101].*

the plugin FracLac [30] used in FIJI software is very similar (0.96 and 0.99, respectively). Since dislocations are line defects, these values are very reasonable. As the macroscopic strain increases, new dislocations are generated, or, even, subgrains

*Bright field image of a magnesium alloy deformed around 2% in tensile mode at a temperature of 200°C and a*

*(a) Idealized microstructure of a completely dislocation-free grain before plastic deformation. (b–d) Subgrain formation during the deformation process. (e) Subgrain coarsening during deformation process.*

**Figure 2.**

**108**

**Figure 1.**

*strain rate of 10*�*<sup>3</sup> s*

�*1*

*Fractal Analysis - Selected Examples*

*. Zone axis [1120] g = [1100].*

*(f) Microstructure shown in (c) with an artifact.*

tilted at two different angles. In the first case (**Figure 5a**), a staking fault is observed, and it is fully parallel to the beam. Therefore, atomic lines are clearly observed. When

The mechanical properties of metals are highly influenced by temperature. In general, the mechanical strength decreases significantly, and the ductility and plastic

T > 0.5Tmelt, time-dependent plasticity becomes dominant [31]. This time-dependent deformation is also called creep. In this temperature range, the dislocations increase their mobility due to the activation of the climbing mechanism, in particular when the staking fault energy is high. The activation of dislocation climbing is revealed in the appearance of well-ordered dislocation structures generated during deformation. In the following section, some examples of dislocation structures generated at room temperature and high temperature (573 K) will be shown. In the second case, the temperature effect on the arrangement of dislocations to form subgrain bound-

*Bright field image of tangled dislocations in pure aluminum deformed up to 22.3% of plastic strain at room*

deformation increases with temperature. In addition to these effects, when

the sample is tilted (**Figure 5b**), it is possible to distinguish the stacking fault throughout the thickness of the TEM sample. Again, the fractal dimension changes.

**4. Temperature effect on dislocation fractal structures**

*Fractal Analysis of Strain-Induced Microstructures in Metals*

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

aries will be observed.

**Figure 6.**

**111**

*temperature at 104*�*s*

�*1 .*

**Figure 4.** *TEM image of a magnesium alloy deformed at 3% at room temperature and a strain rate of 10*�*<sup>3</sup> s* �*1 . Zone axis [1120] g = [1101]. (a) Dark field and (b) bright field.*

conditions where images from one single mode, bright or dark, are enough for a complete description of the microstructure. In particular, after mechanical testing conducted at high temperature, the dislocation substructure is easily observed in bright field mode because the lattice distortion is almost eliminated.

Finally, it is interesting to point out that TEM images have a 2D nature, whereas dislocations, twins, subgrains, and, in general, the crystal structure have a 3D nature. Cui and Ghoniem [13] studied the influence of size on the fractal dimension of dislocation structure throughout dislocation dynamics simulations. The analysis showed that the DF of the 3D structure is significantly smaller than the DF of the 2D corresponding to the projected dislocations in all considered sizes. **Figure 5** shows two images of a 3% deformed magnesium alloy at room temperature of the same area

#### **Figure 5.**

*Bright field image of a magnesium alloy deformed at 3% of plastic strain at room temperature. (a) Sample fully parallel to the beam and (b) sample tilted with respect to the previous condition.*

tilted at two different angles. In the first case (**Figure 5a**), a staking fault is observed, and it is fully parallel to the beam. Therefore, atomic lines are clearly observed. When the sample is tilted (**Figure 5b**), it is possible to distinguish the stacking fault throughout the thickness of the TEM sample. Again, the fractal dimension changes.
