# Research on Corrosion Fatigue Performance and Multiple Fatigue Sources Fracture Process of Corroded Steel Wires

Authors: Ying Wang; Yuqian Zheng
Journal: Advances in Civil Engineering
 (2019)

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## Abstract

Corrosion fatigue (CF) failure is one of the typical failure modes of high-strength steel wires for bridge cables because the cables are subjected to long-term fatigue loads and exposed to heavily polluted environment simultaneously. In this paper, a numerical simulation method was proposed to study CF performance of corroded high-strength steel wires. Firstly, the cellular automata (CA) method was used to generate a numerical model of corroded steel wires with corrosion pit, which can accurately describe the electrochemical process of metal corrosion. In the established CA model, three kinds of cells were involved, namely, metal cell, passive film cell, and corrosive medium cell. By setting 10 cellular transformation rules, morphology of the random corrosion pit on the steel wire surface was simulated. And then, a damage evolution model related to coupling of corrosivemedium and fatigue loads (CCF) was developed to describe the CF damage evolution process of steel wires. Subsequently, the damage evolution process was analyzed by ABAQUS with a user-defined material subroutine (UMAT). Finally, the life of corroded steel wires was predicted, and the CF performance of corroded steel wires with multiple corrosion pits was evaluated. The results show that the proposed method can reasonably describe the CF damage evolution process and illuminate the failure mechanism of steel wires subjected to the CCF. Damage of the steel wire with a single corrosion pit evolves gradually, and the damage evolution rate increases. For the steel wires with multiple corrosion pits, the corrosion pits affect mutually in the fracture process. When the angle and distance between corrosion pits reach a certain degree, the mutual effects can be ignored. With the same pit depth, the angle and distance among corrosion pits determine the CF life of steel wires mainly, and the number of corrosion pits affects slightly.


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## Body

## 1. Introduction

Cable-supported bridge is one of the main structural types in modern bridges, including the suspension bridge, the cable-stayed bridge, and the arch bridge. Due to its strong spanning ability, graceful appearance, and construction convenience, the cable-supported bridge has been widely used in the projects of cross-river/sea bridges all over the world. Considering the loading characteristics of the cable-supported bridge, the cable system is the most important component of the bridge and controls the service life of the whole bridge [1]. Because the cables are subjected to long-term cyclic loads and exposed to heavily polluted environment at the same time, the cable system is vulnerable. Within the past 30 years, there were many incidents caused by the damage and breakage of cables [2–4]. Correspondingly, plenty of rehabilitation works of the deteriorated bridge cable were necessarily carried out [5, 6]. The service life of these damaged cables is less than 20 years and seems too short compared with the 100-year to 120-year design service life of bridges.With considering its comprehensive mechanical properties, such as high strength, good plasticity, and good toughness, hot-dip galvanized high-strength steel wire is the most widely used material for bridge cables [7]. Generally, a sound steel wire possesses a strong ability to resist the fatigue crack nucleation but has little ability to prevent the fatigue crack propagation [8]. When the surface of steel wire is corroded, its mechanical properties will be degraded significantly, which has a great influence on its service life. Many researchers carried out uniaxial tensile tests on the corroded steel wire and analyzed the effects of corrosion on the elastic modulus, the yield strength, the ultimate strength, and the ultimate strain of steel wire. Some static performance degradation models of steel wire were proposed [9–11]. Some fatigue tests [12, 13] of corroded steel wire also were tackled. The results show that even a tiny pitting defect may involve into a main source of fatigue crack nucleation. Meanwhile, the antifatigue crack nucleation ability of steel wire degrades obviously. The fatigue life of the corroded steel wire decreases significantly. Nakamura and Suzumura [14] considered that the fatigue performance degradation of the corroded steel wire was mainly caused by the stress concentration around the surface defects. Therefore, based on various experiments, researchers carried out a lot of research work on the service performance deterioration of steel wire, which was the basis of life prediction of steel wires for bridge cables.The above works, however, are confined to solving practical engineering problems and only focus on the fatigue properties of corroded steel wire. The works on the corrosion fatigue (CF) properties of steel wire are limited. In fact, CF failure is one of the typical failure modes of high-strength steel wires for bridge cables because the bridge cables are subjected to long-term cyclic loads and exposed to heavily polluted environment at the same time. The CF failure is a coupling process of metal brittle fracture under periodic/cycle or nonperiodic/random alternating load and corrosive media. Its essence is the interaction of the electrochemical corrosion process and mechanical process [15, 16]. CF failure possesses many special characteristics. Different from the mechanical fatigue, fatigue limit for the CF failure does not exist. The CF failure may occur under the circumstance of low stress [17]. The CF failure also is different from the static stress failure and may occur in every metal material under any corrosive condition [18]. In addition, unlike the precorrosion fatigue, CF failure is a not simple or sequential superposition of corrosion and fatigue, but the mutual promotion and direct coupling of these two effects [19]. Therefore, further study on the CF performance of steel wire is necessitated.This paper aims to develop a numerical method to describe and visualize the damage evolution of high-strength steel wire under the coupling of corrosion and fatigue (CCF). Typically, under CCF, the CF damage evolution process usually originates from a corrosion pit after the passivation film is broken and mainly includes two stages of CF crack nucleation and CF crack propagation, as shown in Figure1 [20]. Hence, the first step of CF damage simulation is to establish a numerical model of steel wire with an initial corrosion pit. The cellular automata (CA) method is used to generate the position and morphology of the random corrosion pit. By using different kinds of cells and setting certain transformation rules, the CA method can simulate the dynamic process of metal corrosion and electrochemical reaction. Then, a user-defined material subroutine (UMAT) is compiled to embed the constitutive equation of damage evolution of steel wire in the commercial software ABAQUS. Finally, the cyclic block-based method is utilized to realize the simulation of the CF damage evolution process of steel wire, and the damage evolution process is visualized. With the proposed simulation method, the CF damage evolution is evaluated and the CF life is predicted for steel wires with single or multiple corrosion pits.Figure 1
Schematic diagram of the CF damage evolution process [20].


## 2. CA Simulation of Metal Corrosion Process

Electrochemical inhomogeneity in the metal surface often leads to the electrochemical corrosion and then gives rise to the formation of the corrosion pit. Corrosion pit generating stress concentration finally results in the CF crack nucleation, which is called the process of pit accelerating crack nucleation [21]. Typically, the pit growth and pit-crack transition account for more than 90% of the CF life of steel wire [22], which are related with the distribution and morphology of the initial corrosion pit. Hence, establishing the initial corrosion pit model is the key issue to study the CF life with the theory of pit accelerating crack nucleation. However, due to the influence of material characteristics and environmental factors, simulating the metal corrosion process is cumbersome to keep in line with the actual condition. Taking an accurate prediction of the distribution and morphology of the corrosion pit is very difficult. At present, some uncertain methods such as neural network model [23], probabilistic analysis process model [24, 25], and stochastic process model [26] are often used in literatures to evaluate damage evaluation and to predict life. Based on these methods, some key variables such as environmental and electrochemical factors are taken into account in the corrosion process to establish the nonlinearity mapping relations between these variables and the corrosion eigenvalues (the mass loss rate, the pit depth, and the pit shape factor). A more reliable pit model should include not only physical and chemical factors but also a large number of random parameters from solid mechanics, electrochemistry, and materials science.In this paper, the CA method [27] is adopted to generate corrosion pits in corroded steel wire randomly. Different kinds of cells and specific cellular transformation rules could be used conveniently to simulate the dynamic process of metal corrosion. Cellular is the most basic constituent element of the CA system. By setting a series of cellular transformation rules, the transformation of different cellular types can be realized. The corrosion process involves at least three materials, namely, metal, passive film, and corrosion medium. Therefore, in the CA space, these three materials are discretized into metal cell, passive film cell, and corrosive medium cell. In order to simulate the CF behavior of cylindrical steel wire, these cells are arranged in a three-dimensional cylindrical CA space in order and have their own position. According to the basic electrochemical reaction in the corrosion process, cellular transformation rules are defined to realize the dynamic process of metal corrosion in a mesoscopic level. The current position of one type of cell may be occupied by another type of cell of the neighboring position. This rule of occupation is based on the electrochemical equation. Finally, the distribution and morphology of the initial corrosion pit are formed randomly.When the oxide film on the metal surface is damaged, metal in a humid environment will undergo electrochemical reaction. The basic equation of metal dissolution can be expressed as follows [28]:(1)Me+H2O⟶MeOHaq+1/2H2where MeOH4 is the substance in solution after the reaction, which has no influence on the subsequent corrosion process. Therefore, in the CA model, the reaction of metal dissolution can be expressed as follows:(2)M+C⟶Cwhere M is the metal cell and C is the corrosion cell. With this rule, after the metal cell reacts with the corrosion cell, the corrosion cell that can further corrode the metal cell will be formed. In the CA space, the current position of the cell M is occupied by the cell C of the neighboring position and then the corresponding neighboring position becomes empty.On considering the geometry of steel wire, the cylindrical coordinate system is selected. The length of the wire is 14 mm, and its radius is 3.5 mm. The wire is placed in a corrosion solution with a radius of 7 mm. In order to ensure high solution accuracy, the steel wire and the external corrosion environment are divided into an array consisting of cells where the cell size is 0.1 mm, 0.1 mm, and 3° along the radial, axial, and circumferential direction in the CA model. Each cell is represented by its position coordinate (i, j, and k). Constant boundary condition is selected; that is, the outer cells of the boundary are endowed with a constant to describe a finite CA space. A three-dimensional cell space is established, as shown in Figure 2.Figure 2
Schematic diagram of CA model for the corrosion process of high-strength steel wire.In the CA space of metal corrosion, three cell types are mainly involved (as shown in Figure3):(1)
Metal cell (M): the metal cell can be etched away by the corrosive cell, and its position remains unchanged throughout the simulation.(2)
Passive film cell (P): in the CA space, the passive film cell will cover the outermost metal cell initially. Position of the passive film cell is basically fixed. When the initial defect occurs randomly in the simulation process, the position of the passive film cell will be occupied by a metal cell.(3)
Corrosive cell (C): the corrosive cell is randomly distributed in solution. It can corrode the metal cell and move freely. In the cell space, C can randomly select to move along one direction of the coordinate axis. Here, the concentration of the corrosive solution is 0.1, which means the ratio of the number of corrosive cells to the grid number in the space of corrosive media. The corrosive cells are uniformly distributed.Figure 3
Types of cells in the CA model for the corrosion process of high-strength steel wire.A key issue of the CA system is to define the reasonable cellular transformation rules, which should be able to reflect the real electrochemical corrosion process. When an electrochemical reaction occurs, the state and position of a cell at the next time step will depend on the states and positions of this cell and all of its neighboring cells at the current time step. Here, the CA model containing 6 neighbor cells is adopted, as shown in Figure4, where the dark cells represent the central cell and the light cells represent its neighbor cells.Figure 4
Neighboring cells in the CA model for the corrosion process of high-strength steel wire.Then, cellular transformation rules are defined to simulate the dynamic process of metal corrosion. Three aspects need to be considered: (1) random diffusion of corrosive media in solution; (2) breakage of the passive film on the metal surface; and (3) the electrochemical reaction of the exposed metal surface in a humid environment and the gradual dissolution of metal.For the diffusion process in solution, a cell C can selectively move to one of its neighbors within a time step. If the selected neighbor cell is not occupied by other cell C or cell P, the cell C jumps to the neighbor cell. Alternatively, it remains in the original position.For the passive film breakage and the initial defect generation process, cell P is randomly converted into a cell M, i.e.,(3)P⟶MIn addition, the following spatial correlation is introduced to constrain the random diffusion process. Within a time step, a cell C selects one of its neighbors. If the selected neighbor cell is empty, the cell C remains at its original position. When a cell M is chosen as a target position by two neighbor cells C simultaneously, each cell of the three cells keeps its original position unchangeably.The transformation rule of metal corrosion can be summarized into the following 10 steps (depicted in Figure5):(1)
Within a time step, a cell C randomly selects one of its neighbors and is ready to move to the neighboring position(2)
If the selected neighboring position in step 1 has been occupied by another cell C, the cell C remains in its original position, as shown in Figure5(a)(3)
If the selected neighboring position in step 1 is empty but another cell C is also ready to move to this neighboring position, the cell C remains in its original position, as shown in Figure5(b)(4)
If the selected neighboring position in step 1 is empty and no other cell C is ready to move to this neighboring position, the cell C moves to the neighboring position, and the original position becomes empty, as shown in Figure5(c)(5)
If the selected neighboring position in step 1 is occupied by a cell M but another cell C is ready to move to this neighboring position, the cell C remains in its original position, as shown in Figure5(d)(6)
If the selected neighboring position in step 1 is occupied by a cell M and no other cell C is ready to move to this neighboring position, the cell M disappears, cell C moves to this neighboring position, and the original position becomes empty, as shown in Figure5(e)(7)
If a cell P breaks at this point of time, the cell P becomes a cell M, as shown in Figure5(f)(8)
If the selected neighboring position in step 1 is occupied by a cell P and the cell C is located outside the metal space, the cell C remains in its original position, as shown in Figure5(g)(9)
If the selected neighboring position in step 1 is occupied by a cell P and the cell C is located inside the metal space but another cell C is also ready to move to this neighboring position, the cell C remains in its original position, as shown in Figure5(h)(10)
If the selected neighboring position in step 1 is occupied by a cell P and the cell C is located inside the metal space and no other cell C is ready to move to this position, the cell P disappears, the cell C moves to a neighboring position, and the original position becomes empty, as shown in Figure5(i)Figure 5
The cellular transformation rule for the corrosion process of high-strength steel wire.By establishing the reasonable CA model including metal cell, passive film cell, and corrosive medium cell and setting the above cellular transformation rules, the three-dimensional coordinate data of steel wire with corrosion pit can be obtained in MATLAB software. By importing these coordinate data of steel wire with corrosion pit generated in MATLAB into AutoCAD software, the grid model of steel wire with pit can be obtained, as shown in Figure6(a)). As the data are generated randomly by using CA technology, it can be found that the grid model of steel wire with corrosion pit is prone to form sharp angles, which will lead to the nonconvergence of the subsequent damage calculation. The study found RhinoResurf, a reverse-engineering plug-in in RHINO 3D NURBS software, can efficiently handle complex grid data and conveniently establish 3D surface model and solid model. So the grid model in AutoCAD is firstly imported into RHINO to generate the surface model with a smooth surface, and then the “auto-create-entity” function in RHINO is used to further generate the solid model, as shown in Figure 6(b). Subsequently, the solid model is stored in IGS format and imported into ABAQUS software to obtain a 3D geometry model of steel wire with corrosion pit, as shown in Figure 6(c). Finally, by defining the material constants and meshing the geometry model, a finite element model of steel wire with corrosion pit is established for subsequent damage analysis, as shown in Figure 6(d).Figure 6
The geometric morphology of corrosion pit and the 3D model of high-strength steel wire: (a) the grid model in AutoCAD software; (b) the solid model in RHINO software; (c) the geometry model in ABAQUS software; (d) the FE model in ABAQUS software.
(a)
(b)
(c)
(d)


## 3. Programming Realization of Material Damage Model

At present, there are mainly two methods to simulate CF damage evolution process: one is based on the fracture mechanics (FM) [29–31] and the other is relied on the continuum damage mechanics (CDM) [32, 33]. In the FM-based method, the corrosion pit is considered to be equivalent to surface cracks, and two models have been proposed: the superposition model [34] and the competition model [35]. The FM-based method can quantitatively present the damage evolution process and provide a theoretical framework for the crack nucleation analysis. But there are many factors in the CF crack propagation process, and the coupling relationship between corrosion and fatigue is relatively complicated, which makes the application of these two models very inconvenient. Therefore, in order to fully study the CCF, this paper refers to the CDM model [36] and compiles a user-defined material subroutine (UMAT) to simulate the CF damage evolution of high-strength steel wire in ABAQUS software. UMAT is a FORTRAN interface program provided by ABAQUS to users to customize material properties. Since there is no damage constitutive model of high-strength steel wire under the CCF in ABAQUS, the UMAT is adopted to define the damage constitutive relationship of steel wires under the CCF in this paper. UMAT program can be called directly by ABAQUS, and the process of damage evolution of steel wire under the coupling action can be visualized in ABAQUS software.In continuum damage mechanics, the macroscopic state variableD is commonly used to describe the distribution, the characteristic, and the evolution process of microstructural defects. In general, the physical meaning of D is the ratio of the reduction of the effective area caused by the microstructural defects to the initial undamaged area, i.e.,(4)D=1−AeA0,=AlA0,where A0 is the initial undamaged cross-sectional area, Ae is the effective cross-sectional area, and Al is the reduction of the effective cross-sectional area caused by the microstructural defects. Obviously, the damage state variable D is a dimensionless scalar parameter. When D equals to 0, there is no damage in material. When D equals to 1, the material is totally damaged without any load bearing capacity.As shown in Figure7 [37], for a straight bar under the uniaxial tension, the effective stress is defined as the ratio of the applied force and the effective cross-sectional area, i.e.,(5)σe=FAe=FA01−D=σ1−D,where σ is the normal stress.Figure 7
Schematic diagram of the damage in a bar specimen under uniaxial tension [37].According to the symmetry of the cylindrical steel wire, it is assumed that the rate of CF damage evolution along each radial direction in the cross section is the same. As shown in Figure8 [36], based on the above assumption, the loss of effective radius Rl caused by CF is same along two arbitrary diameters (R1 and R2). Similar to the definition of D in equation (4), a CF model of high-strength steel wire is established according to the effective radius, i.e.,(6)Dcf=1−ReR=RlR,where R is the initial radius of steel wire without damage, Re is the effective radius of steel wire, and Rl is the loss of effective radius.Figure 8
Schematic diagram of the CF damage model of high-strength steel wire.As shown in equation (6), before calculating the CF damage of steel wire, the loss of effective radius Rl should be obtained first. According to [36], Rl can be divided into Rl1 and Rl2 (Rl=Rl1+Rl2). Rl1 refers to the size of defects caused by CCF, which is the pit depth in the pit growth stage or the crack length in the crack propagation stage. Rl2 refers to the crack length within inward of bridge wires due to the action of cyclic stress where it has not been corroded.As introduced in Section2, there are three stages in the CF damage evolution process of steel wires, including the pit growth, crack propagation, and material breakage. According to [38], the time-varying law of pit depth can be described as follows:(7)xpit=actbc,where ac and bc are model parameters related to the corrosion environment. Therefore, the pit growth rate model can be described as follows:(8)dxpitdt=βcαc1/βcxpit1−1/βc.According to [39], the crack propagation rate model can be described as follows:(9)dxcrackdt=CΔSpxcrackq,where C, p, and q are model parameters related to the material characteristics and the corrosion environment.The critical size of pit-to-crack transition is determined by using the rate competition model [35]. The rate competition model indicates that when the pit evolves to a critical state, the extension force of the CF crack propagation has exceeded the driving force of the pit growth, which results in a transition of the CF process. The rate competition model can be described as follows:(10)dxdtcrack≥dxdtpit,where x is the pit depth or the equivalent crack length and dx/dt is the propagation rate of CF crack length or the growth rate of pit depth.Therefore, the critical size of pit-to-crack transition can be described as follows:(11)xc=βcαc1/βcCΔSpβc/1+βcq−1.The critical time point of pit-to-crack transition can be described as follows:(12)tp=xcαc1/βc.In the stage of pit growth, whenx≤xc and t≤tp, the pit growth rate is greater than the crack propagation rate, i.e.,(13)dxpitdt=βcαc1/βcxpit1−1/βc≥dxcrackdt=CΔSpxq.Hence, according to equation (7), Rl1 can be described as follows:(14)Rl1=αctβc.In the stage of crack propagation, whenx≤xc and t≤tp, the pit growth rate is smaller than the crack propagation rate, i.e.,(15)dxpitdt=βcαc1/βcxpit1−1/βc<dxcrackdt=CΔSpxq.Hence, according to equation (9), Rl1 can be described as follows:(16)Rl1=xc1−q1−q+CΔSpt−tp1−q1/1−q.According to equation (4), the reduction of the effective cross-sectional area caused by the cyclic load can be described as follows:(17)Al=DfA0.Hence, the crack length only caused by the cyclic load can be described as follows:(18)Rl2=DfR,where Df is the damage variable of steel wire under the cyclic load. According to [40], Df can be described as follows:(19)Dft=1−1−μ+1ΔSM0Btf1/μ+1,where μ, M0, and B are model parameters and f is the loading frequency of cyclic load.In conclusion, the CF damage evolution modelDcft developed in this paper is(20)Dcft=RlR=Rl1+Rl2R=Df+αctβcR,t≤tp,Df+1−qxc1−q/1−q+CΔSpt−tpR1/1−q,t>tp.In this paper, UMAT compiling with the above CF damage evolution model is used to set the material properties of high-strength steel wire in ABAQUS software. In addition, the step is the basis of the solution process of ABAQUS software, i.e., a step represents a loading or unloading process. Therefore, the UMAT written in this paper is also based on the step, and its algorithm is shown in Figure9.Figure 9
The algorithm flow diagram of UMAT.The UMAT consists of four state variables, from STATEV(1) to STATEV(4). The STATEV(1) and STATEV(2) are used to store the stress of elements in loading steps and that in unloading steps, respectively. The stress is calculated by Jacobian MatrixDDSDDE based on the elasticity modulus E that updates step by step by using the calculated damage variable D. The elasticity modulus E and the damage variable D are stored in STATEV(3) and STATEV(4), respectively.As shown in Figure9, the UMAT mainly includes three parts. The function of each part is to calculate Jacobian Matrix DDSDDE, stress, and strain whose calculation methods are same in each part and then to update all of the state variables. However, the difference of each part is the calculation method of the elasticity modulus E and the damage variable D.The first part represents the condition of the first loading when the step equals 1. At this moment, no damage is accumulated in the material; i.e., the damage variableD is equal to 0 and the elastic modulus E is equal to the initial elastic modulus E0.The second part represents the condition of unloading when the step is even. The unloading step is to make the load change from maximum value to the minimum value and calculate the stress at the end of the step. At this moment, still no damage is accumulated in the material; i.e., the elastic modulusE and the damage variable D are the same as those of the previous step (the loading step), which have been stored in the STATEV(3) and STATEV(4), respectively.The third part represents the condition of loading when the step is odd but not equal to 1. The loading step is to increase the load from the minimum value to the maximum value, calculate the damage variableD, and calculate the stress at the end of the step. Before calculating the damage variable D, it is necessary to calculate the stress difference between the previous loading and unloading step, that is, the stress range ΔS. Using the calculated stress range ΔS, the damage increment ΔD is calculated based on equation (20). The damage increment ΔD refers to the damage produced after a cyclic load as follows: (21)ΔDN=DΔSN−1,N−1−DΔSN−1,N−2,where N is the number of the loading step, ΔDN is the damage increment produced after the previous cyclic load, and ΔSN−1 is the stress difference between the previous loading and unloading step. The increment ΔD is then added into the damage variable D that has been stored in the STATEV(4). The calculated damage variable D refers to the accumulated damage before the current step. It should be noted that, in the second loading step, the damage variable D can be directly calculated because the damage increment ΔD after the first cyclic load is just the damage variable D. If the calculated damage variable D is greater than 0 in the current step, the element has started to accumulate damage. When the calculated damage variable D is greater than a certain value close to 1 in the current step, the element has failed. Regarding the idea of the life-and-death element method, it is necessary to constrain the inactive degrees of freedom in the “dead” element in the solution process, which can reduce the number of solution equations and also prevent position errors. Therefore, the calculated damage variable D needs to be verified to ensure that it is not less than the damage variable D in the previous cyclic loading and is not greater than 0.9. Then, the elastic modulus E is updated according to the damage variable D, i.e.,(22)E=E0×1−D.In order to verify the correction of the compiled UMAT, the result from a uniaxial tensile fatigue test of undamaged steel wire is used [41]. Galvanized high-strength steel wires with a diameter of 7 mm and a length of 150 mm were employed as specimens in the test. The specified ultimate tensile strength of steel wires, commonly used worldwide in bridge cables, was 1670 MPa. During the fatigue test, the stress ratio R was set as 0.5, and four stress ranges were selected as 335 MPa, 418 MPa, 520 MPa, and 670 MPa, respectively. The MTS Landmark 100 kN fatigue testing machine was used for the test, and the loading frequency was 10 Hz. The median S-N curve of steel wire was obtained:(23)lgN=23.480−6.649×lgΔS.The S-N curve is used to fit the parameters in equation (19), and thus, the three coefficients, μ, M0, and B, are equal to 1, 3772.53, and 6.649, respectively. The fatigue damage model in equation (19) is used to compile UMAT-F, and a steel wire model without initial damage is developed, as shown in Figure 10. The density of high-strength steel wire is 7.8 g/cm3, the initial elasticity modulus E0 is 212000 MPa, and Poisson’s ratio is 0.31 [42]. The S-N curve of the steel wire by using the developed UMAT is shown in Figure 11. From Figure 11, the simulation results are compared with the experimental results [41], which show that the developed UMAT in this paper can achieve the expected goal, and the simulation result is highly consistent with the experimental result.Figure 10
The geometry model of the undamaged steel wire.Figure 11
The S-N curve of the undamaged steel wire under the uniaxial tensile fatigue loads.


## 4. Structural Solution Based on the Cyclic Block

In general, the high-strength steel wire used in bridge cables has a long service time. Accumulated damage in the steel wire under a load cyclic is relatively low. Hence, it is time-consuming and also unnecessary to calculate the damage in each cyclic load. Therefore, a concept of cyclic block is introduced; i.e., a cyclic block represents a certain number of cyclic loads. That is to say, within a cyclic block, the damage of material remains unchanged. When analysis conducts to the next cyclic block, the damage of material will be updated according to the state variables of the current cyclic block. Correspondingly, the relationship between the number of cyclic blockNcb and the total number of cyclic loads N is(24)Ncb=intNn,where n is called the computational accuracy, which represents the number of cyclic loads within a cyclic block.However, in the commercial software ABAQUS, the above damage calculation process is based on the analysis steps, which include loading steps and unloading steps. When the number of all previous analysis steps is odd, the current analysis step is the loading step. Otherwise, when the number of all previous analysis steps is even, the current analysis step is the unloading step. The relationship between the number of analysis stepsNas and the number of cyclic blocks Ncb is as follows:(25)Ncb=Nas+12,whenNasis odd,Nas2,whenNasis even.Since the damage value remains unchanged within a cyclic block, then times of loading and unloading cycles within a cyclic block can be equivalent to one loading analysis step and one unloading analysis step. In conclusion, the relationship among the cyclic block, the cyclic load, and the analysis step is as shown in Figure 12.Figure 12
The relationship between the cyclic block, the analysis step, and the number of cyclic loads.
(a)
(b)When steps are set in ABAQUS software, there is only one increment in each step; i.e., the load increases directly from the minimum value to the maximum value or decreases directly from the maximum value to the minimum value at one time. The damage variableD is involved in the structural solution only at the beginning of each cyclic block and has an impact on the performance of the material in each cyclic block. Therefore, within each cyclic block, the structural solution can be regarded as a linear elastic structural solution. However, for the whole analysis process, the structural solution in different cyclic blocks is not completely independent. The accumulated damage in the previous steps needs to be constantly calculated to reduce the structural stiffness. In this way, the material nonlinear behavior caused by accumulated CF damage is presented in the structural solution process.


## 5. CF Fracture Simulation of Steel Wire with the Single Fatigue Source

The parameters of the CF damage model in equation (20) are fitted by using the S-N curve of steel wire obtained from the CF experiment in the literature [1], which are listed in Table 1. The experimental object is the presplit steel wire in [1]. In the CF experiment of steel wire, the corrosion environment is 3.5% NaCl solution, the stress ratio R is −1, the stress ranges ΔS are 500 MPa, 600 MPa, and 800 MPa, and the loading frequency f is 6 Hz.Table 1
Parameter values in the CF damage model.
Model parameter
μ
B
M
0
α
c
β
c
C
q
p
Value
1
6.649
3772.53
1.27
×
10
−
5
0.689
1.8099
×
10
−
9
0.798
2.01In this section, in order to compare the simulation result with the experimental result in the literature [1], the pit depth is adopted as the corrosion eigenvalue to control the corrosion degree. Therefore, in the CA simulation process, the initial corrosion pit depth is set as 0.5 mm. The generated geometry model and FE model of steel wire are shown in Figure 13. In the ABAQUS software, the origin of the global coordinates coincides with the center of the cylinder bottom. The environment and loading conditions are the same as those in the CF experiment. Based on the CF damage model in equation (20), the UMAT-CF is compiled to simulate the damage evolution process of steel wire under the coupling of corrosion fatigue in ABAQUS software. The simulation result of the CF damage evolution process in the steel wire is as follows.Figure 13
The FE model of high-strength steel wire with an initial corrosion pit.Figure14 gives the CF damage evolution process of high-strength steel wire in Figure 13 under the cyclic load of ΔS=600MPa and 3.5% NaCl solution, in which 6 steps are selected, including Step 61, 121, 181, 211, 219, and 221. In each step, three figures are included. The first figure is the stress contours of the middle cross section of steel wire; the second figure is the damage contours of the middle cross section of the steel wire; the third figure shows the damaged elements whose color is blue in the middle cross section, which presents the size of the defect (the pit depth or the crack length).Figure 14
CF damage evolution and stress distribution of high-strength steel wire (view cut:z = 7 mm).
(a)
(b)
(c)
(d)
(e)
(f)The simulation object is a steel wire model with a random corrosion pit, as shown in Figure13. The corrosion environment is 3.5% NaCl environment. The stress ratio R is −1, the stress ranges ΔS is 600 MPa, and the loading frequency f is 6 Hz. Without additional statement, the environment and loading conditions of the following figures are consistent with these figures.From the stress contours in Figure14, the maximum stress appears at the bottom of the pit and the stress distribution in the middle cross section of steel wire gradually changes with the damage evolution process. This is because the elastic modulus E of the material around the corrosion pit will gradually reduce due to the stress concentration, which leads to the local change of the deformation and the decline of the local stiffness. However, the damaged material in the corrosion pit is still able to transfer stress and it is still continuous with the surrounding material, which makes the stress increase in the material in the adjacent area of the corrosion pit. From the damage contours in Figure 14, the damage evolves gradually at the bottom of the pit where the maximum stress appears. There are more and more damage elements, and some elements inside the steel wire are also damaged.In addition, in this paper, the corrosion pit is equivalent to the surface crack, and thus there is no specific distinction between the pit and crack. The pit depth and the crack length are collectively referred to as the defect size. In order to unify the damage assessment system, the stress intensity factorKI is used to quantify the overall damage degree of high-strength steel wire. When the stress intensity factor KI reaches to the fracture toughness KIC, the steel wire ruptures, which is described as follows:(26)KI=Yσπx<KIC,where σ is the stress within the steel wire and Y is the crack shape factor of the semielliptic surface crack in the cylinder that is subjected to tension loads, which can be expressed as follows [14]:(27)Y=1.84/πtanπx/2D/πx/2D1/2cosπx/2D×0.752+2.02x/D+0.371−sinπx/2D3,where x is the depth of corrosion pit and D is the radius of steel wire. According to [7], the fracture toughness KIC of steel wire with the radius of 7 mm is about m.Figure15 shows the time-varying curves of the stress intensity factor KI and the size of defect x of steel wire model in Figure 13 under the cyclic load of ΔS=600MPa and 3.5% NaCl solution. As shown in Figure 15, with the increase in loading step, the damage gradually accumulates. Correspondingly, the defect size x and the rate of damage evolution gradually increases. When the step equals 101, 201, 301, 369, and 371, the defect size x is 0.80547, 0.83748, 1.19706, 2.00606, 2.3587, and 2.46322 mm, respectively. The rate of defect growth in each loading step interval above is 6.402×10−4, 7.192×10−3, 2.697×10−2, 8.816×10−2, and 1.045×10−1 mm/cyclic block, respectively. When the step equals 369, the defect size x is 2.3587 mm, and hence, the stress intensity factor KI is 64.6602MPam that is near to the fracture toughness KIC. It means that the steel wire ruptures in the 369th step.Figure 15
The time-varying curves of the stress intensity factorKI and the size of defect x. The original data of the table and figures in this section are provided in S1 File of the section “Supplementary Materials.”Based on the above method, the CF life of steel wires under cyclic loads with different stress ranges is calculated, and the results are shown in Table2. According to the CF life in Table 2, the S-N curves are drawn in Figure 16. From Figure 16, compared with the S-N curve obtained from the experiment [1], the life of wire obtained by simulation seems to be significantly higher than that obtained by the experiment, especially when the stress level is relatively low. The main two reasons caused this difference are as follows: (1) the different initial defects; in the experiment, the initial defect is a presplit crack (0.5 mm in depth) processed by digital technology, while in the simulation, the initial defect is a random corrosion pit (0.5 mm in depth) generated by the CA method. The growth capacity of crack is obviously stronger than that of pit. (2) the different damage evolution processes; the experiment only studied the crack propagation stage of steel wire whose surface is damaged, while the simulation studied the whole process from pit growth, pit-to-crack transition, and crack propagation to the steel wire rupture. Therefore, it is reasonable that the simulation result is higher than the experimental result. In addition, considering the complexity of CF process and the randomness of the defect generation process, it is difficult to design an experiment that is totally in line with the simulation process. Hence, it is concluded that the simulation method proposed in this paper can basically reach the expected goal and has strong reliability and practicability for predicting CF life of steel wires with initial random corrosion pit and simulating CF damage evolution.Table 2
CF life of high-strength steel wire with a corrosion pit.
Δ
S
MPa
Computational accuracy
Step
x
mm
K
I
MPa
m
The number of cyclic loads
Simulation CF life
Experiment CF life [1]
Error (%)
500
500
61
0.74527
19.280
15,500
58,000
37,071
36.1
121
0.80547
20.0380
30,500
181
1.0427
24.6642
45,500
221
1.9159
42.5406
55,500
231
2.6528
62.7575
58,000
233
2.8601
69.8541
58,500
600
200
101
0.7453
23.1366
10,200
37,000
19,142
48.3
201
0.8366
25.1065
20,200
301
1.1970
33.0457
30,200
361
2.0058
53.6176
36,200
369
2.3410
64.0653
37,000
371
2.4403
67.4662
37,200
800
100
61
0.7992
32.3990
3,100
11,300
6,968
38.3
121
0.9929
38.0034
6,100
181
1.3565
48.9707
9,100
221
1.7703
62.7539
11,100
225
1.7990
63.7790
11,300
227
1.8587
65.9452
11,400
The original data of the table and figures in this section are provided inS1 File of the section “Supplementary Materials.”Figure 16
CF S-N curves of steel wires obtained from simulation and experiment.The experiment object is a steel wire with a presplit crack in [1]. Both in experiment and simulation, the corrosion environment is 3.5% NaCl environment. The stress ratio R is −1, the stress ranges ΔS is 500 MPa, 600 MPa, and 800 MPa, and the loading frequency f is 6 Hz. The original data of the table and figures in this section are provided in S1 File of the section “Supplementary Materials.”


## 6. CF Fracture Simulation of Steel Wires with Multiple Corrosion Pits

The steel wire with a single fatigue source is the mainly discussed object in Section5, as shown in Figure 17(a). From Figure 17(a), the CF fractography of this kind of steel wire can be divided into three zones, namely, crack initiation zone, crack propagation zone, and instantaneous fracture zone [8]. That is to say, the fatigue crack initiates in a certain pitting defect on the surface, namely, the fatigue source of crack initiation. The crack propagation zone is the zone extending from the center of the fatigue source to the periphery, which appears very smooth after repeated friction under the high-frequency fatigue load. The instantaneous fracture zone is the zone where the fracture occurs when the crack propagates to the critical size, which has an irregular-stepped shape [8]. But beyond that, there are two fracture types of steel wires with multiple fatigue sources. One is known as the fracture with planar multiple fatigue sources, which means that the initiation and propagation of crack are generally in the same plane, as shown in Figure 17(b). Another one is known as the fracture with stepped multiple fatigue sources, which means that initiation and propagation of cracks are not in the same plane, as shown in Figure 17(c).Figure 17
SEM scanning diagram of the CF fracture surface of steel wires [8]: (a) the single fatigue source; (b) the planar multifatigue sources; (c) the stepped multifatigue sources.
(a)
(b)
(c)At present, most of simulation studies on CF fracture only focus on the single fatigue source and do not consider the interaction between the multiple corrosion pits [33, 36]. Hence, the conclusions and results obtained from these studies are inadequate. In order to study the fracture process of steel wires with the multiple fatigue sources, in this section, multiple pits with the pit depth of 0.5 mm are set up in the steel wires, and the influence of the number and distribution of pits on the CF life of the steel wires are studied.


### 6.1. CF Fracture with Planar Multiple Fatigue Sources

Seven steel wires with different corrosion morphologies are analyzed, as shown in Figures18 and 19. Case A is the steel wire with a single pit that has been discussed in Section 5. The location of the pit is shown in Figure 18(a), and the model of the corroded steel wire is shown in Figure 19(a). Case B, Case C, and Case D are the steel wires with two pits in the same cross section perpendicular to the z-axis, in which the angles between two pits are 60°, 90°, and 120°, respectively. The locations of the pits are shown in Figures 18(b)–18(d), respectively, and the models of the steel wires are shown in Figures 19(b)–19(d), respectively. Case E, Case F, and Case G are the steel wires with three pits in the same cross section perpendicular to the z-axis, in which the angles between pits are 60°, 90°, and 120°, respectively. The locations of the pits are shown in Figures 18(e)–18(g), respectively, and the steel wire models are shown in Figures 19(e)–19(g), respectively. Other model parameters are the same as those in Section 5.Figure 18
Schematic diagram of the cross section of steel wires with the different planar multifatigue sources: (a) Case A; (b) Case B; (c) Case C; (d) Case D; (e) Case E; (f) Case F; (g) Case G.
(a)
(b)
(c)
(d)
(e)
(f)
(g)Figure 19
Steel wire model with the different planar multifatigue sources (view cut:z = 7 mm): (a) Case A; (b) Case B; (c) Case C; (d) Case D; (e) Case E; (f) Case f; (g) Case G.
(a)
(b)
(c)
(d)
(e)
(f)
(g)The UMAT-CF is used to calculate the CF life of the steel wires under a cyclic load with a stress rangeΔS of 600 MPa. The other environment and loading conditions are the same as those in Section 5. The calculation results are shown in Table 3. Comparing the results of Case A, Case B, and Case E corresponding to the conditions of one pit, two pits, and three pits, the influence of number of pits on the CF life of steel wires seems unobvious and is discussed specifically as follows.Table 3
CF life of high-strength steel wires with different planar multifatigue sources.
Case
The number of corrosion pits (num)
The angle between corrosion pits (θ)
CF life (Ncf)
Case A
1
—
37,000
Case B
2
60
∘
37,000
Case C
2
90
∘
38,000
Case D
2
120
∘
38,200
Case E
3
60
∘
37,400
Case F
3
90
∘
38,200
Case G
3
120
∘
38,400
The original data of the table and figures in this section are provided inS2 File of the section “Supplementary Materials.”Figure20 presents the time-varying curves of the stress intensity factor KI of steel wires under CCF for the Case A, Case B, and Case E. From Figure 20, at the initial stage of damage evolution, the CF damage evolution rate enhances with the increase in the number of corrosion pits. However, when the steel wires are broken, the CF damage evolution rate reduces with the increase in the number of corrosion pits. Therefore, the CF life of steel wires under the three different cases is basically the same.Figure 20
Time-varying curves of the stress intensity factorKI of steel wires corresponding to the cases in Figure 19. The original data of the table and figures in this section are provided in S2 File of the section “Supplementary Materials.”The points of A, B, and C in Figure21 are corresponding to the damage distribution of steel wires under CCF for the Case A, Case B, and Case E when the analysis step is equal to 301. From Figure 21(a), at the initial stage of damage evolution, the damage zones around the corrosion pits on the steel wire models evolve independently. At this moment, the mutual influence between the corrosion pits is negligible. In addition, according to the study of Hu et al. [33], the level of stress concentration is closely related to the morphology of corrosion pit. Therefore, the following conclusion can be obtained: the initial damage evolution rate is mainly related to the morphology of initial corrosion pits, but the initial corrosion pits in this paper are randomly generated by the CA method, so the initial damage evolution rate is also uncontrollable and is irrelevant to the number of corrosion pits. The points of D, E, and F in Figure 21 are corresponding to the damage distribution of steel wires under CCF for the Case A, Case B, and Case E when the step is equal to 369. From Figure 21(b), at the late stage of damage evolution, the damage zones around the corrosion pits on the steel wire models for Case B and Case E have affected each other, thereby accelerating the damage evolution rate. In summary, at the late stage of damage evolution (the steel wires are going to fracture), the number of planar multiple fatigue sources will affect the CF damage evolution rate, but the influence is not significant, which results in that the CF life of steel wires will not change with variation in the number of corrosion pits.Figure 21
The damage cloud map of steel wires corresponding to the points in Figure20: (a) step = 301; (b) step = 369.
(a)
(b)From Table3, comparing the results of Case B, Case C, and Case D corresponding to the conditions with different angles between two pits, the greater the angle between pits is, the longer the CF life of steel wires is. Figure 22 presents the time-varying curves of the stress intensity factor KI of steel wires under CCF for the Case B, Case C, and Case D. From Figure 22, the CF damage evolution rate of steel wires decreases generally as the angle between pits increases. More specifically, the gap between the damage evolution process of Case C and that of Case D is not significant, but the damage evolution rate of these two cases is smaller than the evolution rate of Case B.Figure 22
Time-varying curves of the stress intensity factorKI of steel wires corresponding to the cases in Figure 19. The original data of the table and figures in this section are provided in S2 File of the section “Supplementary Materials.”The points of A, B, and C in Figure23 are corresponding to the damage distribution of steel wires under CCF for Case B, Case C, and Case D when step equals 331. From Figure 23, at the 331st step, the damage zones around the corrosion pits for Case B has already penetrated and are much greater than those for Case C and Case D. At this moment, for Case C and Case D, the damage zones around the corrosion pits still evolve independently because of the relatively long distance between the two pits. Hence, the slight gap of CF life between these two cases may be caused by the initial morphology of corrosion pits. It can be included that, for the fracture process with multiple fatigue sources, the corrosion pits will affect each other, but when the angle between pits increases to a certain extent, their mutual influence can be ignored, which is consistent with the conclusion in [43].Figure 23
Damage cloud map of steel wires corresponding to the points in Figure22.
(a)
(b)
(c)


### 6.2. CF Fracture with Stepped Multiple Fatigue Sources

In this section, five steel wires with different corrosion morphologies are analyzed, as shown in Figures24 and 25. Case A is the steel wire with a single pit that has been discussed in Section 5. The location of the pit is shown in Figure 24(a), and the model of the corroded steel wire is shown in Figure 25(a). Case B, Case C, and Case D are the steel wires with two pits in the same cross section perpendicular to the x-axis, in which the distances between two pits are 1 mm, 2 mm, and 4 mm, respectively. The locations of the pits are shown in Figures 24(b)-24(d), respectively, and the models of the steel wires are shown in Figures 25(b)–25(d), respectively. Case E is the steel wire with three pits in the same cross section perpendicular to the x-axis, in which the distance between pits is 2 mm. The locations of the pits are shown in Figure 18(e), and the steel wire model is shown in Figure 19(e). Other model parameters are the same as those in Section 5.Figure 24
Schematic diagram of the cross section of steel wires with the different stepped multifatigue sources: (a) Case A; (b) Case B; (c) Case C; (d) Case D; (e) Case E.
(a)
(b)
(c)
(d)
(e)Figure 25
Steel wire model with the different stepped multifatigue sources (view cut:x = 0 mm): (a) Case A; (b) Case B; (c) Case C; (d) Case D; (e) Case E.
(a)
(b)
(c)
(d)
(e)The UMAT-CF is used to calculate the CF life of the steel wires under a cyclic load with a stress rangeΔS of 600 MPa. The other environment and loading conditions are the same as those in Section 5. The calculation results are shown in Table 4. Comparing the results of Case A, Case C, and Case E corresponding to the conditions of one pit, two pits, and three pits, the CF life of steel wires seems positively correlated with the number of pits. The reason for this phenomenon is discussed specifically as follows.Table 4
CF life high-strength steel wires with different stepped multifatigue sources.
Case
The number of corrosion pits (num)
The distance between corrosion pits (d) (mm)
CF life (Ncf)
Case A
1
—
37,000
Case B
2
1
37,400
Case C
2
2
38,000
Case D
2
4
38,400
Case E
3
2
38,400
The original data of the table and figures in this section are provided inS3 File of the section “Supplementary Materials.”Figure26 presents the time-varying curves of the stress intensity factor KI of steel wires under CCF for the Case A, Case C, and Case E. From Figure 26, the gap of CF life between these three cases mainly results from the different evolution rates of the initial damage; i.e., the KI of Case A is markedly greater than those of Case C and Case E.Figure 26
Time-varying curves of the stress intensity factorKI of steel wires corresponding to the cases in Figure 25. The original data of the table and figures in this section are provided in S3 File of the section “Supplementary Materials.”Figure27 presents the damage distributions of steel wires under CCF for the Case A, Case C, and Case E when step equals 331, which are corresponding to the points of A, B, and C in Figure 26. From Figure 27, for steel wire models with multiple fatigue sources, the damage zones around the corrosion pits on the steel wire models evolve independently. When the damage around one of the pits reaches the state of fracture, the whole steel wires rupture. This process is consistent with the damage evolution process of steel wires with a single corrosion pit. In short, the number of corrosion pits in steel wires with stepped multifatigue sources has little effect on the CF damage evolution process, and the difference in CF life between Case A, Case C, and Case E is mainly caused by the randomness of the initial morphology of corrosion pits.Figure 27
Damage cloud map of steel wires corresponding to the points in Figure26.
(a)
(b)
(c)Comparing the results of Case B, Case C, and Case D corresponding to the pit distance of 1 mm, 2 mm, and 4 mm in Table4, the CF life of steel wires seems positively correlated with the distance between two corrosion pits. The reason for this phenomenon is discussed specifically as follows.Figure28 presents the time-varying curves of the stress intensity factor KI of steel wires under CCF for the Case B, Case C, and Case D. From Figure 28, the damage evolution rate at the initial stage of damage evolution is independent of the distance between the pits. But as the damage accumulates, the damage evolution rate increases gradually. At this moment, the evolution rate is negatively correlated with the distance between the pits, which leads to that the CF life of steel wires enhances as the distance between the pits increases.Figure 28
Time-varying curves of the stress intensity factorKI of steel wires corresponding to the cases in Figure 25. The original data of the table and figures in this section are provided in S3 File of the section “Supplementary Materials.”Figure29(a) presents the damage distributions of steel wires under CCF for the Case B, Case C, and Case D when step equals 325, which are corresponding to the points of A, B, and C in Figure 28. From Figure 29(a), at the initial stage of damage evolution, the damage degree is mainly related to the morphology of corrosion pits and the damage zones around each pit evolve independently, so the damage evolution rate is independent of the distance between the pits. Figure 29(b) presents the damage distributions of steel wires under CCF for the Case B, Case C, and Case D when step equals 375, which are corresponding to the points of D, E, and F in Figure 28. From Figure 29(b), when step equals 375, the damage zones around the corrosion pits have penetrated for Case B and Case C. However, at this moment, since the distance between the two corrosion pits is too large, the damage zones around the corrosion pits are independent mutually and still remain independent when the steel wire is going to rupture. Therefore, it can be concluded that, for the fracture process with stepped multiple fatigue sources, the damage degree is mainly related to the morphology of the corrosion pits at the initial stage of damage evolution, and damage evolves independently around corrosion pits. As the damage zones gradually expand, the corrosion pits will affect each other. However, when the distance between pits increases to a certain extent, their mutual influence can be ignored, which is also consistent with the conclusion in [43].Figure 29
Damage nephogram of steel wires corresponding to the points in Figure28: (a) step = 325; (b) step = 375.
(a)
(b)


## 6.1. CF Fracture with Planar Multiple Fatigue Sources

Seven steel wires with different corrosion morphologies are analyzed, as shown in Figures18 and 19. Case A is the steel wire with a single pit that has been discussed in Section 5. The location of the pit is shown in Figure 18(a), and the model of the corroded steel wire is shown in Figure 19(a). Case B, Case C, and Case D are the steel wires with two pits in the same cross section perpendicular to the z-axis, in which the angles between two pits are 60°, 90°, and 120°, respectively. The locations of the pits are shown in Figures 18(b)–18(d), respectively, and the models of the steel wires are shown in Figures 19(b)–19(d), respectively. Case E, Case F, and Case G are the steel wires with three pits in the same cross section perpendicular to the z-axis, in which the angles between pits are 60°, 90°, and 120°, respectively. The locations of the pits are shown in Figures 18(e)–18(g), respectively, and the steel wire models are shown in Figures 19(e)–19(g), respectively. Other model parameters are the same as those in Section 5.Figure 18
Schematic diagram of the cross section of steel wires with the different planar multifatigue sources: (a) Case A; (b) Case B; (c) Case C; (d) Case D; (e) Case E; (f) Case F; (g) Case G.
(a)
(b)
(c)
(d)
(e)
(f)
(g)Figure 19
Steel wire model with the different planar multifatigue sources (view cut:z = 7 mm): (a) Case A; (b) Case B; (c) Case C; (d) Case D; (e) Case E; (f) Case f; (g) Case G.
(a)
(b)
(c)
(d)
(e)
(f)
(g)The UMAT-CF is used to calculate the CF life of the steel wires under a cyclic load with a stress rangeΔS of 600 MPa. The other environment and loading conditions are the same as those in Section 5. The calculation results are shown in Table 3. Comparing the results of Case A, Case B, and Case E corresponding to the conditions of one pit, two pits, and three pits, the influence of number of pits on the CF life of steel wires seems unobvious and is discussed specifically as follows.Table 3
CF life of high-strength steel wires with different planar multifatigue sources.
Case
The number of corrosion pits (num)
The angle between corrosion pits (θ)
CF life (Ncf)
Case A
1
—
37,000
Case B
2
60
∘
37,000
Case C
2
90
∘
38,000
Case D
2
120
∘
38,200
Case E
3
60
∘
37,400
Case F
3
90
∘
38,200
Case G
3
120
∘
38,400
The original data of the table and figures in this section are provided inS2 File of the section “Supplementary Materials.”Figure20 presents the time-varying curves of the stress intensity factor KI of steel wires under CCF for the Case A, Case B, and Case E. From Figure 20, at the initial stage of damage evolution, the CF damage evolution rate enhances with the increase in the number of corrosion pits. However, when the steel wires are broken, the CF damage evolution rate reduces with the increase in the number of corrosion pits. Therefore, the CF life of steel wires under the three different cases is basically the same.Figure 20
Time-varying curves of the stress intensity factorKI of steel wires corresponding to the cases in Figure 19. The original data of the table and figures in this section are provided in S2 File of the section “Supplementary Materials.”The points of A, B, and C in Figure21 are corresponding to the damage distribution of steel wires under CCF for the Case A, Case B, and Case E when the analysis step is equal to 301. From Figure 21(a), at the initial stage of damage evolution, the damage zones around the corrosion pits on the steel wire models evolve independently. At this moment, the mutual influence between the corrosion pits is negligible. In addition, according to the study of Hu et al. [33], the level of stress concentration is closely related to the morphology of corrosion pit. Therefore, the following conclusion can be obtained: the initial damage evolution rate is mainly related to the morphology of initial corrosion pits, but the initial corrosion pits in this paper are randomly generated by the CA method, so the initial damage evolution rate is also uncontrollable and is irrelevant to the number of corrosion pits. The points of D, E, and F in Figure 21 are corresponding to the damage distribution of steel wires under CCF for the Case A, Case B, and Case E when the step is equal to 369. From Figure 21(b), at the late stage of damage evolution, the damage zones around the corrosion pits on the steel wire models for Case B and Case E have affected each other, thereby accelerating the damage evolution rate. In summary, at the late stage of damage evolution (the steel wires are going to fracture), the number of planar multiple fatigue sources will affect the CF damage evolution rate, but the influence is not significant, which results in that the CF life of steel wires will not change with variation in the number of corrosion pits.Figure 21
The damage cloud map of steel wires corresponding to the points in Figure20: (a) step = 301; (b) step = 369.
(a)
(b)From Table3, comparing the results of Case B, Case C, and Case D corresponding to the conditions with different angles between two pits, the greater the angle between pits is, the longer the CF life of steel wires is. Figure 22 presents the time-varying curves of the stress intensity factor KI of steel wires under CCF for the Case B, Case C, and Case D. From Figure 22, the CF damage evolution rate of steel wires decreases generally as the angle between pits increases. More specifically, the gap between the damage evolution process of Case C and that of Case D is not significant, but the damage evolution rate of these two cases is smaller than the evolution rate of Case B.Figure 22
Time-varying curves of the stress intensity factorKI of steel wires corresponding to the cases in Figure 19. The original data of the table and figures in this section are provided in S2 File of the section “Supplementary Materials.”The points of A, B, and C in Figure23 are corresponding to the damage distribution of steel wires under CCF for Case B, Case C, and Case D when step equals 331. From Figure 23, at the 331st step, the damage zones around the corrosion pits for Case B has already penetrated and are much greater than those for Case C and Case D. At this moment, for Case C and Case D, the damage zones around the corrosion pits still evolve independently because of the relatively long distance between the two pits. Hence, the slight gap of CF life between these two cases may be caused by the initial morphology of corrosion pits. It can be included that, for the fracture process with multiple fatigue sources, the corrosion pits will affect each other, but when the angle between pits increases to a certain extent, their mutual influence can be ignored, which is consistent with the conclusion in [43].Figure 23
Damage cloud map of steel wires corresponding to the points in Figure22.
(a)
(b)
(c)


## 6.2. CF Fracture with Stepped Multiple Fatigue Sources

In this section, five steel wires with different corrosion morphologies are analyzed, as shown in Figures24 and 25. Case A is the steel wire with a single pit that has been discussed in Section 5. The location of the pit is shown in Figure 24(a), and the model of the corroded steel wire is shown in Figure 25(a). Case B, Case C, and Case D are the steel wires with two pits in the same cross section perpendicular to the x-axis, in which the distances between two pits are 1 mm, 2 mm, and 4 mm, respectively. The locations of the pits are shown in Figures 24(b)-24(d), respectively, and the models of the steel wires are shown in Figures 25(b)–25(d), respectively. Case E is the steel wire with three pits in the same cross section perpendicular to the x-axis, in which the distance between pits is 2 mm. The locations of the pits are shown in Figure 18(e), and the steel wire model is shown in Figure 19(e). Other model parameters are the same as those in Section 5.Figure 24
Schematic diagram of the cross section of steel wires with the different stepped multifatigue sources: (a) Case A; (b) Case B; (c) Case C; (d) Case D; (e) Case E.
(a)
(b)
(c)
(d)
(e)Figure 25
Steel wire model with the different stepped multifatigue sources (view cut:x = 0 mm): (a) Case A; (b) Case B; (c) Case C; (d) Case D; (e) Case E.
(a)
(b)
(c)
(d)
(e)The UMAT-CF is used to calculate the CF life of the steel wires under a cyclic load with a stress rangeΔS of 600 MPa. The other environment and loading conditions are the same as those in Section 5. The calculation results are shown in Table 4. Comparing the results of Case A, Case C, and Case E corresponding to the conditions of one pit, two pits, and three pits, the CF life of steel wires seems positively correlated with the number of pits. The reason for this phenomenon is discussed specifically as follows.Table 4
CF life high-strength steel wires with different stepped multifatigue sources.
Case
The number of corrosion pits (num)
The distance between corrosion pits (d) (mm)
CF life (Ncf)
Case A
1
—
37,000
Case B
2
1
37,400
Case C
2
2
38,000
Case D
2
4
38,400
Case E
3
2
38,400
The original data of the table and figures in this section are provided inS3 File of the section “Supplementary Materials.”Figure26 presents the time-varying curves of the stress intensity factor KI of steel wires under CCF for the Case A, Case C, and Case E. From Figure 26, the gap of CF life between these three cases mainly results from the different evolution rates of the initial damage; i.e., the KI of Case A is markedly greater than those of Case C and Case E.Figure 26
Time-varying curves of the stress intensity factorKI of steel wires corresponding to the cases in Figure 25. The original data of the table and figures in this section are provided in S3 File of the section “Supplementary Materials.”Figure27 presents the damage distributions of steel wires under CCF for the Case A, Case C, and Case E when step equals 331, which are corresponding to the points of A, B, and C in Figure 26. From Figure 27, for steel wire models with multiple fatigue sources, the damage zones around the corrosion pits on the steel wire models evolve independently. When the damage around one of the pits reaches the state of fracture, the whole steel wires rupture. This process is consistent with the damage evolution process of steel wires with a single corrosion pit. In short, the number of corrosion pits in steel wires with stepped multifatigue sources has little effect on the CF damage evolution process, and the difference in CF life between Case A, Case C, and Case E is mainly caused by the randomness of the initial morphology of corrosion pits.Figure 27
Damage cloud map of steel wires corresponding to the points in Figure26.
(a)
(b)
(c)Comparing the results of Case B, Case C, and Case D corresponding to the pit distance of 1 mm, 2 mm, and 4 mm in Table4, the CF life of steel wires seems positively correlated with the distance between two corrosion pits. The reason for this phenomenon is discussed specifically as follows.Figure28 presents the time-varying curves of the stress intensity factor KI of steel wires under CCF for the Case B, Case C, and Case D. From Figure 28, the damage evolution rate at the initial stage of damage evolution is independent of the distance between the pits. But as the damage accumulates, the damage evolution rate increases gradually. At this moment, the evolution rate is negatively correlated with the distance between the pits, which leads to that the CF life of steel wires enhances as the distance between the pits increases.Figure 28
Time-varying curves of the stress intensity factorKI of steel wires corresponding to the cases in Figure 25. The original data of the table and figures in this section are provided in S3 File of the section “Supplementary Materials.”Figure29(a) presents the damage distributions of steel wires under CCF for the Case B, Case C, and Case D when step equals 325, which are corresponding to the points of A, B, and C in Figure 28. From Figure 29(a), at the initial stage of damage evolution, the damage degree is mainly related to the morphology of corrosion pits and the damage zones around each pit evolve independently, so the damage evolution rate is independent of the distance between the pits. Figure 29(b) presents the damage distributions of steel wires under CCF for the Case B, Case C, and Case D when step equals 375, which are corresponding to the points of D, E, and F in Figure 28. From Figure 29(b), when step equals 375, the damage zones around the corrosion pits have penetrated for Case B and Case C. However, at this moment, since the distance between the two corrosion pits is too large, the damage zones around the corrosion pits are independent mutually and still remain independent when the steel wire is going to rupture. Therefore, it can be concluded that, for the fracture process with stepped multiple fatigue sources, the damage degree is mainly related to the morphology of the corrosion pits at the initial stage of damage evolution, and damage evolves independently around corrosion pits. As the damage zones gradually expand, the corrosion pits will affect each other. However, when the distance between pits increases to a certain extent, their mutual influence can be ignored, which is also consistent with the conclusion in [43].Figure 29
Damage nephogram of steel wires corresponding to the points in Figure28: (a) step = 325; (b) step = 375.
(a)
(b)


## 7. Conclusions

This paper proposed a simulation method to study the CF damage evolution of high-strength steel wires for bridge cables and predicting their CF life. The main conclusions of this study could be drawn as follows:(1)
In order to simulate the actual metal corrosion process, the shape and position data of the initial random corrosion pit on the steel wire surface are obtained by the CA method, and the geometry model of steel wire with the corrosion pit is established. The developed model does not need to presuppose the shape and the position of the initial pitting defect and can fully consider the randomness of the metal corrosion process caused by the electrochemical reaction. The distribution of the corrosion pit can be determined visually on the steel surface.(2)
A numerical simulation method for the CF damage evolution process of steel wire is proposed. Based on the continuum damage mechanics, an UMAT is compiled by FORTRAN. Based on the concept of life-and-death element method, the simulation of the CF damage evolution process is realized. With the developed method, the predicted CF life of steel wires is a little higher than the experimental results. The main reason caused this difference is that the prefabricated defect in the simulation is a random corrosion pit, while the prefabricated defect in the experiment is a crack. These two kinds of defects have different stress concentration degrees. Therefore, the simulation method proposed in this paper can basically achieve the expected goals and has strong reliability and practicability to predict CF life and simulate CF damage evolution of steel wires with an initial defect.(3)
During the CF damage evolution process, the stress distribution in the cross section of steel wire changes gradually, and the stress concentration occurs at the bottom of the corrosion pit. While the damage gradually evolves, there are more and more damaged elements, and some of the elements inside the steel wire are also damaged.(4)
The influence of the number and distribution of the multiple corrosion pits on the CF performance and the CF life of steel wires is studied under the circumstance of the same initial pit depth. The results show that, in the fracture process of steel wires, multiple corrosion pits will affect each other. However, when the angle or distance between the pits increases to a certain extent, the mutual influence between the pits can be neglected. Under the circumstance of the same pit depth, the angle and the distance among corrosion pits could influence the CF life of steel wires positively, but the number of corrosion pits has slight influence on it.



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*Source: 4914359-2019-12-10.xml*