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Fractal Effect of Corrosion on Mechanical Behavior of Unprotected Structural Steel

Currently, it is known that, most of the time, the approach to continuous media results are unsatisfactory for use in real materials. In man made structures, defects appear during the production stage at nano-, microand macro-scales, which evolve throughout its useful life, generating failures, some of which can be catastrophic. Fractography surveys show the nonEuclidean nature of fracture patterns [1]. Macro-scale breaking patterns are common in experimental laboratory research, while microand nano-scale processes, which govern the macroscopic behavior and deformed solids fracture, are not [2].

structural applications. Unfortunately, there were also many accidents leading to human and material losses due to failure of such structures. Some of them were due to defective designs, but it has also been discovered that the defects in materials like persistent cracks might have started cracking, thus leading to fracture. The surface of materials depends a lot on properties such as: adhesion, friction, wear, permeability, etc. It is true that surfaces exposed to the environment and interactions between both of them are responsible for the behavior of materials. Data that might be obtained from the surface may be related to microstructure and mechanical properties of material. It must also be remembered that when a metallic piece is under stress, corrosion concentrates causing transgranular or intergranular cracks.
In the specific case of surface fracture, its analysis is usually carried out through metallography, which may tell us about, among others things, the origin of the fracture, the direction of its propagation and the type of load that caused it. The surfaces of fractures are usually a set of repeated patterns, for example, fractures, fissures, ridges, holes and intergranular defects, among others. Quantitative fractography aims at translating such features to a parametrical shape, and that is where fractal geometry comes in [5].
Fractal geometry looks for regularity in the relationship of an object and its parts at various scales, that is, it studies invariable geometrical aspects, regardless of scale changes. In other words, fractal geometry studies non-differentiable or non-continuous geometrical shapes at any scale. From a mathematical point of view, a fractural is a subset of a metric space for which its Hausdorff-Besicovitch dimension is strictly higher than its topological dimension. In general, fractals have some type of similarity and it may be considered that they are made of small parts similar to the whole. This similarity may be strictly geometrical, or only approximate or statistical. Mandelbrot quantitatively explored for the first time, the fractal character of fracture surfaces subject to various thermal treatments and reported a correlation between fractal dimension and tenacity of the fracture, their property estimated through impact energy [4]. These results, though questioned in the beginning, started a new era in the fractography disciplines. From then, analysis of self-affinity of fracture surface is a very active research field, which has had the contribution of modern and sophisticated statistical and mathematical methods.
There are various methods for analyzing self-affinity of fracture surfaces, most of them use profiles extracted from surfaces through the use of experimental techniques. Highly refined experiments, focused on various materials, do not confirm correlation between the fractal dimension and mechanical properties [6]. It has been established that rugosity exponent ζ is the proper parameter by which to describe fracture surfaces in fast kinetic conditions analyzed mainly through scanning electron microscope (SEM). Bouchaud proposed a universal rugosity exponent ζ = 0.78, regardless of microstructure and properties. Such universality was questioned by the discovery of another self-affine regime characterized by a rugosity exponent ζ = 0.5 for fracture surfaces generated in slow crack propagation conditions or analyzed at manometric scales, using atomic force microscope (AFM). Coexistence of both regimes in various materials has also been reported [7]. Such regimes cross in a fracture length, which seems to be independent of kinetic conditions. Attempts to relate such fracture length with microstructural parameters of some materials continue to be carried out. [8] The development of this research on behavior of unprotected A36 steel exposed to corrosion (the steel had no coating technique applied using zinc as the main protective element), resulted from observing occurrence of cracks on structures exposed to a marine environment, as well as the analysis of consequences implied by such fact, regarding marine corrosion. Research is shown from dimensioning of test specimens; laboratory tests with their chemical analysis; and, finally, a fractography of the corresponding test specimen, obtaining its fractal dimension based on the theoretical development shown.

Dimensioning of test specimens
In accordance with ASTM A-370 standard specifications [9] and complying with such parameters (geometrical relationships between length, width and height), Figure 1 shows the type of test specimen used. Table 1 shows proportions for dimensioning of the test specimen used in the laboratory, obtained in accordance with ASTM standards. Test specimens were made of A36 structural steel (C = 0.29 max., Mn = 1.20, Si = 0.40 max., P = 0.40 max., S = 0.05 max). In accordance with ASTM specifications, test specimen were made with A36 structural steel, chemical composition: C = 0.29 max., Mn = 1.20, Si = 0.40 max., P = 0.40 max., S = 0.05 max. Existing Mexican regulations were considered for tensile tests on steel products. [10]

Laboratory tests and chemical analysis
Tensile tests were carried out in a 250 KN Universal Mechanical Testing machine. Conditions for the test were: testing speed: 0.60 mm/min.; sampling frequency: 5.0 points/s; downwards. Traction assays for unprotected A36 structural steel were carried out on three types of sample, taking into account the base material, and other two samples were applied under real marine corrosion, throughout 6-month and 1-year aging periods, Figures 1.a., 1.b and 1.c. Based on features and components of the metal used, the sample was roughed down and polished in order to remove scratches. The procedure used was the traditional one, consisting in roughing down the surface of the sample and sanding it with an ultimate approximation to a flat surface free from scratches with a moisturized spinning wheel. The sample was moisturized in a Petri dish with a solution of Nital dissolved in water for 10 seconds. Moisturizing time was carefully controlled. Action of moisturizing stopped when the sample was put under a water stream, cleaned with alcohol and a drier was used for finishing [11].

Results of mechanical properties of A36 structural steel
A fractographic analysis begins with a final observation of the fracture's surface features. Evidence of nucleation of fissure, mechanism and direction of propagation causes may be obtained, and an estimation of acting loads' magnitude may eventually be obtained. [12] and [13] Figure 2.a shows the results of the stress-strain graph for virgin material (i.e. not exposed to marine conditions), for which a maximum 369.18522 MPa load and a 255.1545 MPa yield strength were obtained. Figure 3.a shows its fractography. For structural steel aged more than 6 months at sea, a 351.50238 MPa maximum load and a 238.676 MPa yield strength were obtained, Figure 2.b. Fractography in Figure 3.b shows nucleated cracks and a behavior ductile to fracture. For structural steel aged more than 1 year, a 315.883 MPa maximum load and a 226.24 MPa yield strength were obtained, Figure 2.c. Fractography shown in Figure 3.c shows crack branching due to tensile testing, since material loses ductility due to corrosion. The second and third columns of Table 2 show the results of the mechanical properties of steel in accordance with the tensile test, and the fourth and fifth columns show specific values of steel regarding the Internet web page referred to [14]. It should be noticed that results obtained from tensile test for virgin A36 structural steel test specimen are very similar to those shown by that Web page.
Stress under which the material changes its behavior from elastic to plastic is not easily detected, so for this reason the yield strength has been determined drawing a line parallel to the starting section of the stress-strain curve, but displaced 0,002 mm/mm (0,2 % from the origin), crossing the curve of the diagram. The intersection point is the yield point.
To obtain fractal dimension of microcracks found in the material, the theoretical description of a fractal's behavior in unprotected A36 steel exposed to corrosion is shown [15].

FRACTAL EFFECT OF CORROSION ON MECHANICAL BEHAVIOR OF UNPROTECTED STRUCTURAL STEEL
Francisco Casanova-del-Angel

A fractal theoretical desciption
Definition of generating curve. Let I 0 be a unitary lenght line segment, contained in a closed interval, that is, I 0 ⊂ [a, b]. Let I 1 be a set with sectioned behavior, consisting in three segments of a straight line which create, based on starting point a of I 0 , two scalene triangles reflected regarding the middle point c of I 0 , obtained as follows: the first half of segment I 1 is substituted or removed by the sides of triangle which create an angle with I 0 . This process is repeated for the second half, but with the sides reflected from middle point c.
This process is known as generator which is called state 1. Construction of set I 2 is made applying the generator to every segment of I 1 , which is called state 2. Thus, set I k is created applying generator I 1 on every segment of I k-1 , which is called state k, Figure 4.a.

FRACTAL EFFECT OF CORROSION ON MECHANICAL BEHAVIOR OF UNPROTECTED STRUCTURAL STEEL
Francisco Casanova-del-Angel   Somer particular aspects presented in the construction of this type of fractal should be noticed when generator is created by combination or triangular layouts (in size and form): • States I k-1 and I k are different from each other in the sequence shown by poligonal curves ∀ k → ∞, • set I has a fine structure, that is, it contains every detail in every arbitrary small scale.
• although the generator is composed by two triangular figures corresponding to euclidean logic, the geometry of I k ∀ k → ∞ is too irregular to be described in classical geometric terms. Curve I is characterized by being of a similar scale since, based on a transformation F: ℜ n → ℜ n with λ i > 0 ∀ i ∃ n a, b ∈ℜ n such that │F i (a)-F i (b)│ = λ i │a-b│. Similarity of scale is present for triangles created with I 0 , by generator I 1 . In a like manner, it also has the property of being affine, since, based on transformation F already defined, F(a) = T(a) + α with T a non-singular linear transformation and α ∈ ℜ n . It must be remembered that affinity is conceived as a shear transformation or resistant to cutting, and is a contracting or expanding effect, not necessarily in the same direction. A curve type l k satisfies the scale principle if all its relative figures are linked to each other by a scale law.
Let I be a Borel's set such that I = {I 1 , I 2 , I 3 }, where I j is a finite succesion of line segments creating the generator ∀ j = 1, 2, 3, in order that I i = {U 3 j=1 I j i ∀ i = 1, 2, … } is a countable sequence of sets. Thus, measurement μ, of segments I i is defined as: When a geometrical discontinuity is of the fractal type, generated by a natural process, a uniform reticulate should be built. Let (χ, P(χ), μ) be a space with measures such that the sample space is χ = [0, 1] and P(χ) is a set of subsets of χ, where the measurement is μ. Since the system is dynamic, χ ⊆ ℜ p is the phase space. Let us consider a χ reticulate covered by p-dimensional boxes with radius δ n , where B δn (t) is the neighbor box containing the segment of straight line or point t. Succession of neighbor boxes has radius δ n → 0 as n → ∞.
Let us suppose that there is subset I ≠ ϕ from euclidian space n-dimensional, ℜ n , and that |I| = sup{|a -b | such that a, b ∈ I}. If {I i } is a countable set of neighbors with radius δ covering I, thus, there exists a subset II of ℜ n such that II ⊂ (∪∞ i=1 I i ) with |I i | ∈ [0, δ] ∀ i. Therefore, {I i } is neighbor δ of I. If there is a k > 0, then for every neighbor δ > 0 a function to minimze total covering II may be defined as follows: If in (2) we consider the limit, then lím Η k δ (II) = Η k (II). It should be taken into account that Η k (II) is known as Hausdorf's k-dimensional measurement.

Meshing and definition of fractal outline
Let N(t*, Δt*) be the number of squares contained by reticulate, and N(t, Δt) the number of squares intersected by the fractal curve. D θ°( I) shall be the fractal dimension, L the total length of the object, and l the length of every segment. Therefore, L/l quotient defines the number of subdivisions contained by every side of the intersected reticulate. These scale properties correspond to a fragmented fractal, and the multifractal [16], pp. 45 concept is applied. Based on the above: where t ∈ [t*, t*+Δt*]. P(t*) is the probability of distribution of intersection points t ∈ [t*, t* +Δt*] and f(t*) the fractal dimension of such points.
Considering a random generation of f(t), the original curve is rotated to different angles, preferably constant, in order to calculate D θ° for every case.

Behavior pattern
The fractal behavior of a geometrial discontinuity takes us to the concept of diagonal selfaffinity diagonal. In order to define the pattern of the fractal generator, let us begin drawing straight lines from its base to the points where such curves are all along its length, and thus obtain a fractioned curve. It is important to draw horizontal lines in case there is a change in its path behavior, in order to identify the affinity along such path.
Horizontal lines identifying the beginning of the generator, must show the feature of proportionality d such that: L is the length of the generator, l the length of every segment, and L/l is the number of subdivisions contained by the generator. In order to observe the ideal behavior of the path, the scale relationship is defined as the average of the lengths of adjoining generators, that is, (L 1 + L 2 )/2; Figure 2.a.
The scale factor of every rotation undergone by the generator or fractal curve, s, is defined as: where the value of N is the average of the highest number of every rotation, that is: Based on the above, we may get the scale factor of the generator, defined as the inversion of s which, applied, generates the geometrical structure.

Calculation of fractal dimension of the crack for 6-months test specimen
In order to calculate the fractal dimension for cracks generated on test specimen exposed to marine corrosion, both for 6 and 12 months, the theory presented above was applied to fragmented fractals, or fractals generated by natural processes [17]. The original crack was rotated 45°, 90°, 135° and 360° or 0° degrees in order to determine in a more reliable manner the fractal dimension, D f . The process was to choose a picture for the 6-months test specimen and mesh it in order to calculate its fractal dimension, which is shown in Figures 5.a and 5.b.
In order to obtain fractal dimension D f , an average of calculated dimensions for each of the four rotations applied to the fractal curve was obtained. Fractal dimension for a crack on structural steel exposed 6-months to marine conditions is: In order to verify that fractal dimension calculated under the theory developed above is valid, the commercial software Benoit was used. The fractal dimension obtained with such program is D f = 1.34029 [18]. As may be seen, the results are similar.   Figure 6.a shows fractal behavior of the crack, and Figure 6.b shows its digitalization. To obtain fractal dimension of the 1-year test specimen in marine conditions, the same calculation procedure used for the 6-month test specimen was carried out.  Fractal dimension for a crack in A36 structural steel exposed to marine conditions throughout a year is as follows:

Calculation of fractal dimension of the crack for 1-year test specimen
That is, fractal dimension of the cracks in A36 structural steel exposed to marine conditions throughout a year has a 1.3958 value. In accordance with Benoit commercial software, its fractal dimension is 1.40040 [18].

Microanalysis
The X-ray microanalysis study on test specimen aged 12-months at sea shows presence of carbon, oxygen, iron, silica, sulphur, chloride, sodium, calcium and magnesium as the most representative elements, Figure 7. Chloride linked to sodium; basic constituents of sea water, are present as sodium chloride. Magnesium is present in constant relation with chloride which, when combined with other elements, creates magnesium chloride and magnesium sulfate. Sulphur is present in sulphates. All these elements have made possible acceleration of test specimen's corrosion.

Discussion of results
Virgin A36 structural steel showed a ductile fracture surface, which caused concentration of stresses and a local increase of plastic deformation. The elements showed uniform corrosion for both cases of marine exposition, which caused a generalized thinning on the entire surface. They also showed pitting corrosion and a fragile and porous surface.
A loss of ductility of 11.96 % (6 months) and 43.87 % (12 months) were recorded after marine exposition, regarding the virgin test specimen. These small values do not allow creation of a function for loss of ductility due to permanence of steel in Pacific Ocean coasts. This loss of ductility is mainly attributed to the presence of hydrogen, an element present in sea water and organic compounds on the site of marine exposition, causing it to be absorbed and transported to the inside, towards imperfections of the element, thus causing separation of grain borders.
In the case of tensile tests, breaking had to be achieved in order to calculate the fractal dimension of crack generated. The corresponding stress-strain graphical expression shows variation in the fluency strength in each exposition, due to the reduction of the area based on type of uniform corrosion found.
Once the crack was obtained, it was digitalized and the description format to obtain its fractal type was applied. Then, it was verified that such generator was applicable to fractal behavior of the crack. This being so, the crack was meshed, subdivided and rotated at various angles for calculation of its fractal dimension. Fractal dimension calculated shows a ± 0.0036 variation regarding commercial software, which may be due to the box counting algorithm. Fractal dimension obtained for the test specimen exposed 12 months at sea shows a fractal dimension higher than the 6-month test specimen, since the crack has more branches.

Conclusions
Regarding exposition of probes to the environment, it may be affirmed that, due to interaction of an alloy with the environment to which it was exposed, corrosion products were generated on it, which changed failure mechanism of probes subject to tensile assay. Probes exposed on both periods have significant stress concentrations on areas where corrosion developed. This was observed in laboratory assays, since failure appeared in such areas.
Regarding theoretical development of fractals obtained, it has been carried out from line segments creating a generating curve characterized by sets of Borel's theory, which is possible since generation of fractal curve is carried out through natural processes generating geometrical discontinuities of any typical fractal. Characterization of fractal is carried out on a reticulation and its value is based on count of segments for every box in such reticulation. The original crack was rotated 45°, 90°, 135° and 360° or 0° degrees in order to determine in a more reliable manner the fractal dimension, D f . Self-affinity of the fractal is also tested.
Fractal dimension obtained after six months of exposition is lower regarding that obtained at 12 months, which shows that exposure time is directly proportional to cracks generated due to corrosion products and that crack ramification influences the corresponding value.