Computational protocol: Comparing Monofractal and Multifractal Analysis of Corrosion Damage Evolution in Reinforcing Bars

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[…] The specimen used for the accelerated electrochemical corrosion test was about 400 mm in length. shows a schematic representation of the experimental setup. The specimen was immersed into a jug of salt solution with 150 mm height. The detached part of the specimen was coated with insulation tape. The specimen was connected to the anode of a DC power source while the cathode of the DC power source was connected to a copper bar placed parallel to the specimen in the jug. The specimen was periodically washed by using a scale removal solution to remove corrosion products. After weighting, the corrosion mass loss ratio was calculated and the specimen continued to undergo the corrosion process.The specimen used for the accelerated wet-dry cycle corrosion test was also about 400 mm in length and was cut from the same reinforcing bar as that used in the accelerated electrochemical corrosion test. The two end parts of the specimen (each is about 125 mm) were coated with anticorrosive grease and plastic film. The middle part of the specimen (about 150 mm) was designed as the corrosion region, as is shown in . To determine the corrosion characteristics of reinforcing bar in chloride-free and chloride-contaminated simulated concrete solutions, eight specimens were subjected to wet-dry cycle corrosion test. Four specimens were regularly sprayed using the salt solution and the other four specimens were regularly sprayed using the mixture of salt solution and simulated concrete pore solution every 12 hours. When the test was completed, the specimens were washed using the scale removal solution to remove corrosion products. The corrosion mass loss ratio then was calculated.The corrosion morphology images of the specimen were taken by using ME-61 stereomicroscope with magnification of 7X. To avoid the effect of the junction between the corrode and uncorroded parts, only the central region of 140 mm long in the corroded part was taken as the image sampling length. The corrosion morphology images were merged into one picture and then converted to binary images using ImageJ software. Tensile test was also performed for the specimen using standard strength test procedure according to ISO Standards 6892∶1998 to obtain the yield and ultimate strengths of the bar. In the tensile test an electro-hydraulic servo testing machine was used. [...] Fractal dimension is the most important parameter of monofractal theory. Many methods can be used to calculate fractal dimension, among which the box counting method is thought to be particularly suitable for the determination of corrosion morphology. In the box counting method it counts the number of square grids required to entirely cover an object surface, as is shown in . By using different size grids, one can obtain a relation equation between the number of grids and the size of grids. Let ε be the side length of the grid, N(ε) be the number of grids required to cover the corroded area recorded in the image. According to monofractal theory, if an object is fractal, the number of grids and the size of the grids should have the following relationship, (1)where C and D are the constants. D is also called the fractal dimension. In order to explain how to determine these two constants, Eq.(1) is rewritten as follows,(2)By plotting the data set for ln[N(ε)] against ln(ε) and using the least-square fitting method, the fractal dimension parameter D can be easily obtained .In terms of multifractal analysis, it is necessary to define a measure in the digital images which is closely associated with the local corrosion morphology. To calculate the multifractal spectrum, the following definition of measure was used –:(3)where Pij(ε) is the gray value distribution probability in the box(i,j), nij is the gray value of the box(i,j) of size ε. Pij(ε) can be described as multifractal as(4)(5)where the exponent α depending upon the box (i, j) is the singularity of the subset of probabilities, N(ε) the number of boxes of size ε with the same gray value distribution probability, and f(α) the fractal dimension of the α subset. A quantity called partition function, χq(ε), with an exponent τ(q) applied in statistical physics can be constructed as the following equation:(6)where q is the moment order. τ(q) is evaluated by the slope of lnχq(ε)∼ lnε curve. A generalized multifractal spectrum function, f(α), can then be calculated through Legendre transform:(7)Multifractal measures are primarily characterized by their spectrum. The plot of f(α)∼ α is called multifractal spectrum, which is generally a hook-shaped curve. The width of the multifractal spectrum is Δα and the difference of the fractal dimensions of the maximum probability (α = αmin) and the minimum one (α = αmax) is Δf (Δf = f(αmin)-f(αmax)).The generalized dimension, D(q), addresses how mass varies with ε in an image which are calculated from the mass exponent function:(8)For a non- or monofractal the plot of D(q) versus q tends to be horizontal or non-increasing, but for a multifractal, it is generally sigmoidal and decreasing.In general, the related fractal parameters (D and Δα) increase with the increase of the complexity of the corrosion morphology, which can indirectly characterize the corrosion damage of reinforcing bars. For calculating the related fractal parameters, the free plugin FracLac of ImageJ is used. […]

Pipeline specifications

Software tools ImageJ, FracLac
Application Microscopic phenotype analysis
Diseases Nervous System Diseases