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Conceived and designed the experiments: CQ YX. Performed the experiments: YX LP BW CL. Analyzed the data: YX CQ. Contributed reagents/materials/analysis tools: YX LP BW CL. Wrote the paper: YX.

The authors have declared that no competing interests exist.

Based on fractal theory and damage mechanics, the aim of this paper is to describe the monofractal and multifractal characteristics of corrosion morphology and develop a new approach to characterize the nonuniform corrosion degree of reinforcing bars. The relationship between fractal parameters and tensile strength of reinforcing bars are discussed. The results showed that corrosion mass loss ratio of a bar cannot accurately reflect the damage degree of the bar. The corrosion morphology of reinforcing bars exhibits both monofractal and multifractal features. The fractal dimension and the tensile strength of corroded steel bars exhibit a power function relationship, while the width of multifractal spectrum and tensile strength of corroded steel bars exhibit a linear relationship. By comparison, using width of multifractal spectrum as multifractal damage variable not only reflects the distribution of corrosion damage in reinforcing bars, but also reveals the influence of nonuniform corrosion on the mechanical properties of reinforcing bars. The present research provides a new approach for the establishment of corrosion damage constitutive models of reinforcing bars.

The degradation of construction materials due to corrosion is of great concern for national economic development. The most important durability issue with concrete structure is deterioration due to reinforcing bar corrosion

From the damage mechanics point of view, the key in developing the relationship between the macro- and micro-material characteristics is the definition and selection of damage variables, which, at present, has no clear criteria to follow

Based on fractal theory and damage mechanics, the aim of this paper is to describe the monofractal and multifractal characteristics of corrosion morphology and develop a new approach to characterize the nonuniform corrosion degree of reinforcing bars. By comparison, we here propose a multifractal damage variable, which can not only reflect the internal meso-scale corrosion damage but also facilitate the macro-scale analysis of damage mechanics. The relationship between the tensile strengths of the corroded steel bar and the multifractal damage variable is developed using laboratory test results for reinforcing bars of different corrosion levels. The results have important meaning for development of damage mechanics and solution of engineering problems.

Hot rolled plain steel bar (with nominal diameter of 12 mm) according to ISO Standards 6935-1 was used. The nominal carbon concentration is 0.18%. _{2})

White part is ferrite and black part is pearlite.

The specimen used for the accelerated electrochemical corrosion test was about 400 mm in length.

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

The two end parts of the specimen (each is about 125 mm) were coated with solid butter and plastic film and the middle part of the specimen (about 150 mm) was designed as the corrosion region.

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

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 _{ij}(ε) is the gray value distribution probability in the box(i,j), n_{ij} is the gray value of the box(i,j) of size ε. P_{ij}(ε) can be described as multifractal as_{q}(ε), with an exponent τ(q) applied in statistical physics can be constructed as the following equation:_{q}(ε)∼ lnε curve. A generalized multifractal spectrum function, f(α), can then be calculated through Legendre transform:_{min}) and the minimum one (α = α_{max}) is Δf (Δf = f(α_{min})-f(α_{max})).

The generalized dimension, D(q), addresses how mass varies with ε in an image which are calculated from the mass exponent function:

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.

(A) Low corrosion damage degree with slight localize attack, S = 2.4%, (B) High corrosion damage degree with severe localize attack, S = 9.6%.

S (%) | D | Δα | Δf |

2.4 | 1.68 | 1.01 | 0.63 |

4.3 | 1.71 | 1.17 | 0.42 |

6.4 | 1.72 | 1.40 | 0.90 |

9.6 | 1.75 | 1.64 | 0.55 |

12.3 | 1.76 | 1.69 | 0.57 |

As for monofractal analysis, fractal dimension of corrosion morphologies with different corrosion levels is calculated by box counting method. As is shown in

As for multifractal analysis,

In order to demonstrate whether the related fractal parameter can also reflect the corrosion damage in reinforcing bars subjected to different aggressive environments, we calculate the fractal dimension and multifractal spectrums for the eight bars tested using the accelerated wet-dry cycle corrosion method. Four specimens, with code named SC-P1∼SC-P4, were regularly sprayed using salt solution. The other four specimens, with code named SK-P1∼SK-P4, were regularly sprayed using the mixture of salt solution and simulated concrete pore solution. The results, together with the corrosion mass loss ratio obtained from the corrosion test, yield and ultimate strengths obtained from the tensile test, are plotted graphically in

As is shown by

It can be seen from