Three-Dimensional Surface Parameters and Multi-Fractal Spectrum of Corroded Steel

To study multi-fractal behavior of corroded steel surface, a range of fractal surfaces of corroded surfaces of Q235 steel were constructed by using the Weierstrass-Mandelbrot method under a high total accuracy. The multi-fractal spectrum of fractal surface of corroded steel was calculated to study the multi-fractal characteristics of the W-M corroded surface. Based on the shape feature of the multi-fractal spectrum of corroded steel surface, the least squares method was applied to the quadratic fitting of the multi-fractal spectrum of corroded surface. The fitting function was quantitatively analyzed to simplify the calculation of multi-fractal characteristics of corroded surface. The results showed that the multi-fractal spectrum of corroded surface was fitted well with the method using quadratic curve fitting, and the evolution rules and trends were forecasted accurately. The findings can be applied to research on the mechanisms of corroded surface formation of steel and provide a new approach for the establishment of corrosion damage constitutive models of steel.


Introduction
A rough surface with self-similarity and scale invariance always has a fractal geometrical property [1][2]. Many studies have been conducted on the morphology characteristics of rough surfaces on the basis of fractal geometry theory. Many types of fractal surfaces have been established and have played important roles in ultra-precision industries [3][4][5][6], materials science [7][8][9][10], electromagnetic wave scattering [11][12][13], and many other fields. In addition, in actual projects, the morphology of rough surface often presents variability, i.e., anisotropic and local characteristics in spatial distribution [14,15], which makes the fractal geometry theory difficulty to be applied in practical engineering.
As a common building material, steel, because of air-borne chloride ions, moisture, fugitive dust, etc., is highly susceptible to corrosion damage. The surface of corroded steel becomes gradually roughening from the very beginning of a plane. It well known, the surface of corroded steel is mainly roughened by corrosive pitting among all corrosion results, which is generating considerable interest. For geometric morphology, due to a large number of bumps or potholes (pits) and planar regions (without pits) on corroded surface of steel, the rough surface attacked by the corrosive pitting presents larger discreteness and concave convex feature. And the existing experimental data show that the multi-fractal dimension of a corroded steel surface is between two and three. The existing research results show that the surface attack by corrosive damage is characterized by continuity, non-differentiability and self-affinity, and is of multifractal features with a certain measure [16][17][18]. Thus, the research for the multi-fractal features is a powerful tool for analyzing the randomness and discreteness of the damage mechanism of corroded steel. However, the difficulties in the multi-fractal features analysis of corroded steel surface, such as tedious calculations, make it too complex to be applied to practical projects. Therefore, the present study is trying to use a simplified method to study the multi-fractal characteristics of corroded steel surface.
The Weierstrass-Mandelbrot (W-M) method is a common mathematical model used to construct the fractal surface in engineering. In this study, irregular fractal surface of the corroded steel is tried to be constructed by using the W-M method with specific fractal parameters, and calculate its multi-fractal spectrum. Then, based on the shape feature of the multifractal spectrum, the least squares method is applied to quadratic fitting of the multi-fractal spectrum of corroded surfaces. Finally, we conduct quantitatively analysis on the fitting curve to obtain an exact description and accurate analysis of the multi-fractal characteristics of the corroded steel surface.

Material and Specimens
The material used in this study was a Q235 steel, which is a normalized 0.25% low-carbon steel, extensively used in industrial and civil buildings: the matrix consists of a ferritic-pearlitic microstructure with a 20~40μm ferrite grain size, as shown in Fig 1; its basic chemical composition is shown in Table 1. All specimens were cut out from a same steel plate.

Atmospheric Corrosion Experiment
A test sited in Xi'an (latitude 39°45' N, longitude 108°56' E) was selected for the atmospheric corrosion experiment. The test setup of the atmospheric corrosion experiment is shown in Fig 2. During the experimental period, the monthly temperature(T), relative humidity(RH), The Multi-Fractal Spectrum annual precipitation(P), and rainwater acidity(PH) values were measured to identify the atmospheric corrosive environment. According to GB/T 19292.4-2003[19], the specimens for corrosion were 400 mm in length, 60 mm in width and 8 mm in thickness, installed at 45°in the oblique direction. Table 2 presents the atmospheric corrosion conditions measured depending on the test site. After exposed for 0.5, 1, 2 and 4 years, orderly, the specimens were immersed in HCL solution (12% by volume) at room temperature for removing the corrosion products, cleaned with water, dried with hot air, rinsed in acetone, and then kept in a drying oven until the 3D profile measurement [20].

3D Surface Profile Measurements
In order to gain the 3D morphology of the surface of corroded specimens, a non-contact PS50 3D profiler produced by NANOVER Company was used in this study. The 3D profiler was designed with leading edge white light axial chromatism optical pens to obtain nanometer resolution for surface inspection, hi-speed 3D metrology and more precise thickness mapping on a wider range of geometries and materials than any other profiometer [20].The vertical resolution was 0.28μm and the horizontal resolution was 8μm. The setup used for 3D surface morphology measurements is shown in Fig 3A. Each specimen had been detected the corroded surface at a measurement zone of 40 mm×30 mm(along the directions of transverse and longitude, respectively) approximately arranging at the central of specimen, and the scanning step was set to 15μm [20]. The elementary diagram of the measurement area is shown in Fig 3B.

Surface Morphology and Pitting Characterization
Fig 4 clearly shows the surface morphologies of corroded specimens exposed for 0.5, 1, 2 and 4 years, respectively. The images were draw by Golden Software Surfer 8. The software is a fullfunction 3D visualization, contouring and surface modeling package that runs under Microsoft windows, which is used extensively for terrain modeling, bathymetric modeling, landscape visualization, surface analysis, contour mapping, watershed and 3D surface mapping, gridding, volumetric, and much more. In this study, the 3D surface mapping was applied to show more detail of the corroded surface. The profiles of the different measurement regions are presented in Fig 4 (S1 Table). It is well known, the corrosive attack can produce a network of corrosion on metal surface, which may be treated as a single pit or as two or more adjacent pits. Corrosion damage is also related to the link up of adjacent pits during subsequent exposure time. As shown in Fig 4, at the initial stage of corrosion, a few micro-pits only occurred on some particular locations on the corroded surface of specimens, which made the surface slightly irregular (shown in Fig  4A). And with the passage of corrosion time, large amounts of pits occurred on the surface and superimposed each other, which made the corroded surface signally irregular and uneven (shown in Fig 4D). Thus it can be seen that roughness and irregular of the corroded surface increased with the increasing exposure. Although Fig 4 may directly reflect corrosion status of the corroded steel, the quantitative analysis from the angle of mathematics is hard to evaluate. Thus, the engineering application is in urgent need of a method, which can simulate the corroded surface of steel with high-precision and be convenient to quantitatively analyze the simulation.

Construction of the Multi-fractal Spectrum of Corroded Steel Surface
In science and engineering, many mathematical models with different multi-fractal characteristics based on the Weierstrass-Mandelbrot(W-M) method were generally used to analyze the surface conditions [21][22][23][24][25][26]. By using the W-M method, here, we can obtain a fractal surface with respect to an arbitrary surface of corroded steel through setting some specific multi-fractal parameters. For instance, when a fractal dimension (D) was known, the corresponding fractal surface can be obtained by employing the W-M fractal function, and the function is expressed as follows: Where C n is the characteristic length scale, i.e., the scaling constant; n is the wave number; A n and B n , which have uniform distribution in [0, 2π], are independent random number, respectively; D is a multi-fractal dimension between 2 and 3; λ is constant greater than 1.
In this study, the fractal surfaces with D value of 2.2, 2.3, 2.5, and 2.8 were constructed by the W-M method (λ = 1.3) to simulate the surface of corroded steel exposed for 0.5, 1, 2 and 4 years, respectively. Fig 5 shows the W-M surface with different fractal dimensions which were plotted by the MATLAB software.
From Fig 5, it reveals that the fractal surface based on the W-M method was similar with the corresponding surface of corroded steel (shown in Fig 4) in roughness distribution pattern and surface fluctuation trend. Here we evaluated the accuracy of the constructed surface by comparing the measurement data and the theoretical data [27]. where h is the measurement data of practical surface, h ' is the theoretical data of simulated surface, n is the number of date. To obtain the surface height (h or h ' ) of every point, the calculation principle was defined by using the method shown in Fig 6. Noted that the datum plane used to measure the height of the surface was obtained by least squares. Table 3 showed the simulation precisions of fractal surfaces for corroded specimens with different corrosion time. And it can be found that the fractal surfaces with D value of 2.2, 2.3, 2.5, and 2.8 had more than 90% accuracy for simulating the corroded surfaces exposed for 0.5, 1, 2 and 4 years, respectively. That is to say, it is effective to simulate the rough surface of corroded steel by the W-M method.

Calculation of the Multi-fractal Spectrum
Many methods can be used to calculate fractal dimension [28][29][30], among which the multi-fractal dimension approaching to the real value can be obtained by the calculus of variations or the  box counting method. Note that the box counting method, possessing the advantage of clear mathematical principle and simple calculation, was used in this study to calculate the multifractal parameters of corroded steel surface.
A set of boxes with size of l are used to divide the datum plane of the fractal surface of corroded steel [31], make ε = l / L, L = 512 and ε < 1, ν i (ε) is distribution probability of the height of corroded surface for the box(i), which can be calculated as follow: In Eq (3), h i is the height of corroded surface for the box(i), ∑(h i ) j is the sum of the height for all boxes. When the height distribution is of multi-fractal features, it can be described as: M a ðεÞ $ ε Àf ðaÞ ð5Þ where α depending upon the box(i) is the singularity of the subset of height probabilities, M α (ε) refers to the number of boxes having the same height distribution probability when the size of box is ε, and f(α) is the multi-fractal dimension of subset with the α value. Generally, the value of M α (ε) increases with the decreasing of ε [32]. A qth-order partition function applied in statistical physics, χ q (ε), can be described as [33]: tðqÞ ¼ lim where q is the moment order, τ(q) is a non-linear function of q and is known as the mass exponent function. The singularity strength function α and the singularity spectrum f(α) can be calculated through Legendre transform: Theoretically, with the increasing of |q|, the values of α(q) and f(α(q)) are closer to their theoretical limit. But in fact, when q reaches an oversized value, the computational workload will be increased significantly, which will lead to running out of memory; when |q| stands on an undersized value, of which the increment can cause obvious variation of f(α). The reason can be considered as that f(α) calculated through undersized |q| is only part of the multi-fractal spectrum, but cannot completely reflect probability distribution of the fractal surface. In practical, it is impossible to take the value of q to be infinite. But generally, the saturation extent of multi-fractal spectrum increases with the increasing of the q [31]; thus, we determined the value of |q| max through that f(α) and α tend to be saturated. In this study, all of the multi-fractal spectrums of corroded surfaces were obtained with |q| max = 60. Through the analysis above, we calculated f(α)~α of the fractal surfaces (D = 2.2, 2.3, 2.5, and 2.8) corresponding to the corroded surfaces exposed for 0.5, 1, 2 and 4 years.  Table). It can be seen that the shapes of all spectrums are inverted parabolas, but the difference of shapes among the spectrums is still obvious: the multi-fractal spectrum of corroded surface is plumping with the increasing of fractal dimension, and the spectrum span is also widening. It is well known value of Δα (α max − α min ) determines the width of the spectrum, which is the main reason to increase the local indices of the studied variable, i.e., the more the unevenly distributing of surface, the wider the spectrum, the greater the heterogeneity of distribution and vice versa [34]. Thus, the conclusions can also be drawn: the surface of corroded steel unevenly distributes with the increasing of the value of Δα.
Moreover, when the surface of corroded steel is more rough and irregular, the end position of the corresponding multi-fractal spectrum is lower, which means that the proportion of subsets referring to small probability continues to increase for the corroded surface.
As shown in Fig 7, the intersection point of all multi-fractal spectrums is (1.6, 1.6), which indicating the fractal subset with α = 1.6 (the edge of pits) has a largest proportion on the fractal surface. That is to say the fractal subset with α = 1.6 can determine the sharpness or roughness of the corroded steel surface, distinctly. For the Q235 steel under atmospheric corrosion environment, the corrosion damage mainly manifests corrosive pitting due to the heterogeneous microstructures and chemical composition [35,36], as shown in Fig 4. Pits occur at some particular locations of the corroded surface, link up with adjacent pits with the increasing of exposure, and eventually influence the sharpness or roughness of the corroded surface. Thus, it can be seen the effect of atmospheric corrosion on surface condition of Q235 steel can be signally reflected by the development of corrosion pits with the method of multi-fractal.

Fitting of the Multi-fractal Spectrum of Corroded Steel Surface
It can be known from Eq (8) and (9) that the calculation of multi-fractal spectrum is unable to be represented by using a specific or simple function, which makes the multi-fractal spectrum difficult to take quantitative analysis and to be applied in engineering. To simply formulate the multi-fractal spectrums of corroded surface, we had researched their trend and shape, and the following conclusion can be drawn: Each multi-fractal spectrum (shown in Fig 7) can be decomposed into a left part and a right part of inverted parabolas and both of the parts are continuously differentiable.
Therefore, we can fit the multi-fractal spectrums of the fractal surfaces (with D = 2.2, 2.3, 2.5, and 2.8) corresponding to the corroded specimens exposed for 0.5, 1, 2 and 4 years by the method of least squares. And the left and right part of the multi-fractal spectrum can be respectively expressed as piecewise functions as follows for satisfying fitting precision in this study [37]: where α is singularity strength by evaluating a probability of subset, α 0 is the singularity strength for D = D max . A, B, and C are the undetermined coefficients, which can be obtained by a least squares procedure. Herein the absolute value of A is inversely proportional to the value of Δα; the value of C is proportional to the maximum value of f(α). Generally, the greater value of α can reflect a phenomenon that the probability of corresponding subset is smaller; the lower value of α can reflect a phenomenon that the probability of corresponding subset is bigger. The criterion reflected in multi-fractal spectrum can also be explained that the greater value of α can make the small probability of subset to be a significant impact on multi-fractal spectrum; on the contrary, the smaller value of α can make the big probability of subset to be a significant impact. f(α) is a multi-fractal singularity spectrum representing irregularity and complexity of the fractal surface. The extreme values in the distribution of height probability are associated with the low values of f(α), f(α) min and f(α) max , in such a way the big and small probability of height subsets are related to the left and right part of the spectrum, respectively.
Making comprehensive analysis of above discussions, the conclusions can be drawn: the greater the value of C, the higher the f(α) max , and the higher the f(α) max , the more complex and irregular the fractal surface of corroded steel; on the contrary, the smaller the value of C, the lower the f(α) max , and the lower the f(α) max , the more unobvious the complexity and regularity of corroded surface.
Making intuitive observation of each fitting curve in Figs 8-11, |A| of left part curve is more than that of the right part, which explains the phenomenon that the range of the right part of the multi-fractal spectrum is larger than that of the left part. As well known, the larger the range of multi-fractal spectrum, the more the kinds of subsets described by the multi-fractal spectrum. Thus, it can be drawn that the subsets corresponding to the right part of the multifractal spectrum play a leading role in description of the multi-fractal spectrum.
Connecting with corrosion process of steel, when the steel surface is attacked by shallower pits in the initial stage of corrosion, |A| of the right part curve fitting the multi-fractal spectrum is larger and the corresponding range is narrower; the kinds of the subsets included in the corroded steel surface must also be less and the surface is almost a plane. With the increasing exposure time, because the pits contact the sub-surface constituent particles causing further corrosion and link up with adjacent pits damaging the corroded surface, |A| of the right part curve fitting the multi-fractal spectrum must be smaller and the corresponding range is wider; the kinds of the subsets including to the surface of corroded steel must also be more and the surface is more irregular and discrete.
Some parameter of the quadratic fitting is shown in Table 4. Making comprehensive analysis of Table 4 and Figs 4 and 5, the following conclusions can be made that with the increasing of Δα from 1.05 to 1.61 or the decreasing of |A| from 11 to 7, the surface of corroded steel is distributing more unevenly and irregularly; with the increasing of Δf(α) from 0.30 to 0.83, the surface will fluctuate more obviously.
It can be seen from the calculation formula of the multi-fractal spectrum, the weighting factor, q, is the slope of the multi-fractal spectrum, which can also be obtained using: q = @f(q(α)) / @q(α) [38]. For the multi-fractal spectrums mentioned above, whether the right part or the left part, an inequality exists as: According to Eq (19), it can be seen q decreases with the increasing of α. In the left part of the fitting curve, q is a constant positive number; q is a constant negative number in that of the right part. |q| continues to increase with α. When q tends towards the plus infinity, the α min − the minimum value of singularity strength; α max −the maximum value of singularity strength; Δα-D-value of the minimum and maximum value; α 0the singularity strength for D max ; f(α min )-the singularity spectrum for α min ; f(α max )− the singularity spectrum for α max ; f(α) max −the maximum value of singularity spectrum; Δf(α)−D-value of the maximum and minimum spectrum.