Numerical Simulation on Crack–Inclusion Interaction for Rib-to-Deck Welded Joints in Orthotropic Steel Deck

: Weld defects such as porosity, inclusion, burn-through, and lack of penetration are difﬁcult to detect and control effectively in an orthotropic steel deck (OSD), which will be a fatigue crack initiation site and lead to several fatigue cracking. The crack growth behavior in defective welded joints is different from that of defect-free joints. This study investigates crack–inclusion interaction for rib-to-deck welded joints in OSDs based on numerical simulation and linear elastic fracture mechanics (LEFM). A reﬁned ﬁnite element model of a half U-rib with cracks and inclusions was established by using the FRANC3D-ABAQUS interactive technology. The full processes of the crack–inclusion interaction from approaching and penetrating were accurately simulated. Critical parameters, including the stress intensity factor (SIF), the shape factor, the growth rate, and the growth direction were analyzed. The stiff and soft inclusions amplify and shield the SIF of cracks when the crack grows to the local area of inclusions. During the entire process of crack growth, the soft and stiff inclusion accelerate and inhibit the crack growth, respectively. The stiff inclusion will lead to asymmetric growth of the crack shape, where the portion of the crack away from the inclusions has a higher growth rate. The soft and stiff inclusions will attract and repel the direction of crack growth at the proximal point, respectively.


Introduction
Orthotropic steel decks (OSDs) have been widely applied to long-span bridges due to the advantage of lightweight and high load-carrying capacity.However, fatigue cracking is usually detected from the welded joints of a large number of in-service long-span bridges [1][2][3].Most of the fatigue cracks of steel structures initiate from the weld seam.Stress concentration and high residual stress caused by weld defects are the key internal reason for fatigue cracking [4][5][6].While optimization of welding parameters can help reduce weld defects [7,8] and automatic welding technology can improve the welding quality and efficiency of the U-shaped rib unit [9,10], it is important to note that welding defects cannot be completely eliminated.Furthermore, the heavy traffic load can accelerate the cracking of weld defects.The influence of weld defects on the fatigue performance of OSDs deserves investigation.
Significant research progresses have been made on the fatigue cracking of typical welded joints in OSDs.In order to evaluate fatigue reliability, the cumulative damage theory has been widely applied [6,11], and the mechanical indexes have experienced the development history of the nominal stress, the notch stress, the hot spot stress, and the strain energy density.However, due to the inability to consider the crack growth paths, the method is limited in the field of structural safety assessment of weld defects.The fatigue reliability assessment method based on the linear elastic fracture mechanics (LEFM) theory inclusions was established by using the FRANC3D-ABAQUS interactive technology.The full processes of the crack-inclusion interaction from approaching and penetrating were accurately simulated.Critical parameters, such as the SIF, the shape factor, the growth rate, and the growth direction were analyzed.The path-independent integral theory in fracture mechanics is a useful tool for assessing the strength and failure of crack bodies.This theory includes J-integral, M-integral, and L-integral, where M-integral is particularly suitable for evaluating the damage of materials with complex defects or multiple cracks.Yan et al. [29] proposed the M-integral from J-Integral, written as

Theoretical Bases and Numerical Verification
where Γ is the integral path around the crack tip; σ ij is the stress tensor; µ i is the displacement vector; x 1 is the position vector of the integration point; the strain energy density W is defined as W = 1 2 σ ij ε ij , ε ij is the strain tensor; δ 1j is the crack tip opening displacement; q is a function that is 1 at the crack tip and 0 at the boundary of the integration domain, respectively; ds is a small increment along the integration path Γ.
For linear analysis, the filed variables can be obtained by superimposing the actual field with the auxiliary field [29], which is defined as The J-integral after superposition can be expressed as J = J (1) + J (2) + M (1,2)  ( where where M (1,2) is the interaction integral of the auxiliary field and the actual field, i.e., M-integral; A q = L q t ds, q t is the crack front function value, and W (1,2) is the interaction strain energy density, which is defined as In the small yield state, the J-integral is equal to the energy release rate G [30].
where E is elastic modulus and υ is Poisson's ratio; i .The relationship between the M-integral and the SIF K can be written as

I K
(2) In finite element analysis (FEA), the SIFs of different types of fatigue cracks can be solved using the simultaneous Equations ( 4) and (7).

Crack Growth Model
The criterion of fatigue crack growth is the key problem in fatigue cracking analysis.The Paris law model [31] for fatigue crack propagation can be utilized as a connection between the SIF and the crack growth rate, which is written as where N represents the number of cycles; C and m are constants associated with material; ∆K eq represents the equivalent SIF range, ∆K eq = K eq,max − max(K eq,min , 0), 1−υ ; ∆K th represents the fatigue SIF threshold, which indicates that crack does not propagate when ∆K eq < ∆K th .K C represents the fracture toughness, which indicates that the crack will grow rapidly when ∆K eq > K C .
The fatigue life of the crack based on the Paris law can be expressed as where N represents the number of load cycles, where the fatigue crack undergoes from its initial dimension a 0 to the critical dimension a f .

Numerical Verification of FRANC3D-ABAQUS Interactive Technology
The software package FRANC3D (FRANC3D-V8.2.2.2, Fracture Analysis Consultants, lnc., New York, NY, USA) can handle complex structural shapes, load conditions, and crack forms.Unlike other finite elements, FRANC3D uses adaptive mesh re-division technology to ensure a high-quality mesh around the crack tip.The fatigue test results also confirmed the effectiveness and accuracy of FRANC3D in simulating crack growth [32][33][34].In this study, the F-A technology is introduced to analyze crack structures with inclusions.
To evaluate the accuracy and feasibility of the F-A method, a finite wide plate model was established by using ABAQUS (ABAQUS-2021, Groupe Dassult, Paris, France).Semielliptical cracks with varying aspect ratios (a/c) were introduced by FRANC3D, as shown in Figure 1.The SIFs obtained by the F-A method were compared with the theoretical values in the Raju-Newman handbook [35], and the results are presented in Table 1.It is observed that the finite element calculations are in good agreement with the theoretical values, where the relative error is less than 3.0%.This demonstrates the accuracy and feasibility of the F-A method in calculating SIFs for three-dimensional fracture problems.The SIFs obtained by the F-A method were compared with the theoretical values in the Raju-Newman handbook [35], and the results are presented in Table 1.It is observed that the finite element calculations are in good agreement with the theoretical values, where the relative error is less than 3.0%.This demonstrates the accuracy and feasibility of the F-A method in calculating SIFs for three-dimensional fracture problems.

Numerical Simulation Description
An OSD of a suspension bridge with a main span of 820 m [15] was chosen as the engineering prototype for the purpose of simulating the fatigue crack growth characteristics in RTD welds.The dimension parameters of the suspension bridge and RTD are illustrated in Figures 2 and 3a, respectively.The thickness of the deck plate and U-rib are 16 mm and 8 mm, respectively.The upper opening width and height of the U-rib are 300 mm and 280 mm, respectively.The weld penetration rate is 80% for the RTD welding seam.Figure 3b illustrates the typical cracks at welded joints (a joint form of groove welding).The two main types of cracks are the toe-deck crack and the root-deck crack, which initiate from the weld toe and the weld root, respectively.The typical crack will propagate through the deck.This study will focus on the growth characteristics of the toe-deck crack.The process for evaluating growth characteristics of crack interaction with inclusions in RTD welded joints is shown in Figure 4. Initially, a half U-rib model was established by ABAQUS, with an inclusion inserted at the weld toe.Subsequently, the half U-rib model with the inclusion was imported into FRANC3D to generate a local solid model.An initial fatigue crack was inserted near the inclusion on the weld toe line for subsequent analysis.The process for evaluating growth characteristics of crack interaction with inclusions in RTD welded joints is shown in Figure 4. Initially, a half U-rib model was established by ABAQUS, with an inclusion inserted at the weld toe.Subsequently, the half U-rib model with the inclusion was imported into FRANC3D to generate a local solid model.An initial fatigue crack was inserted near the inclusion on the weld toe line for subsequent analysis.
In the model, a combination of the C3D20R hexahedral element and the C3D10 tetrahedral element was used for mesh hybrid division.The non-weld area was divided using the 10 mm hexahedron, while a denser grid of 2 mm was set in the transition section at the weld joint.For the local area containing particle inclusions, a tetrahedral mesh with a 0.5 mm grid was adopted.For the weld area of concern, a 1 mm grid was adopted.Q345qD steel is used in the U-rib and deck plate in the finite element model.The base material has an elasticity modulus of E0 = 206 GPa and Poisson's ratio υ = 0.3.According to the principle of equal strength in steel structure design [36], the seam material is assumed to be the same as the base material.The loading area measures 100 mm × 300 mm, and the stress ratio is 0.1.In the structural analysis, symmetric constraints (Ux = Ry = Rz = 0) were applied at the y-z plane of the deck and U-rib.Additionally, vertical constraints (Uy = 0) were applied on the left deck, and constraints (Uz = 0) were applied on the deck (z = 0) to simulate the actual boundary conditions.As observed from Figure 4, the initial aspect ratio of the crack was assumed to be 0.5 (with a0 = 0.5 mm, c0 = 1 mm), in accordance with the ⅠⅠW [37] recommendation.The crack size limit in the depth direction was assumed half the plate thickness.The inclusion was simulated by spheres with a radius of 1.5 mm.Further research on stress and the propagation of cracks will be conducted based on the FE model.

Crack-Inclusion Stress Distribution
The impact of inclusions on stress distribution around cracks deserves in-depth investigation.It is important to consider the elasticity modulus of different types of inclusions.Sulfide inclusions were categorized as soft inclusions with lower elasticity modulus, whereas oxide inclusions such as MgO and Al2O3 are categorized as stiff inclusions with In the model, a combination of the C3D20R hexahedral element and the C3D10 tetrahedral element was used for mesh hybrid division.The non-weld area was divided using the 10 mm hexahedron, while a denser grid of 2 mm was set in the transition section at the weld joint.For the local area containing particle inclusions, a tetrahedral mesh with a 0.5 mm grid was adopted.For the weld area of concern, a 1 mm grid was adopted.Q345qD steel is used in the U-rib and deck plate in the finite element model.The base material has an elasticity modulus of E 0 = 206 GPa and Poisson's ratio υ = 0.3.According to the principle of equal strength in steel structure design [36], the seam material is assumed to be the same as the base material.The loading area measures 100 mm × 300 mm, and the stress ratio is 0.1.In the structural analysis, symmetric constraints (Ux = Ry = Rz = 0) were Metals 2023, 13, 1402 7 of 17 applied at the y-z plane of the deck and U-rib.Additionally, vertical constraints (Uy = 0) were applied on the left deck, and constraints (Uz = 0) were applied on the deck (z = 0) to simulate the actual boundary conditions.
As observed from Figure 4, the initial aspect ratio of the crack was assumed to be 0.5 (with a 0 = 0.5 mm, c 0 = 1 mm), in accordance with the IIW [37] recommendation.The crack size limit in the depth direction was assumed half the plate thickness.The inclusion was simulated by spheres with a radius of 1.5 mm.Further research on stress and the propagation of cracks will be conducted based on the FE model.

Crack-Inclusion Interaction Analysis 4.1.1. Crack-Inclusion Stress Distribution
The impact of inclusions on stress distribution around cracks deserves in-depth investigation.It is important to consider the elasticity modulus of different types of inclusions.Sulfide inclusions were categorized as soft inclusions with lower elasticity modulus, whereas oxide inclusions such as MgO and Al 2 O 3 are categorized as stiff inclusions with higher elasticity modulus [38].To simplify the analysis, the elasticity modulus of soft and stiff inclusions E 1 are assumed to be 0.25E 0 and 4E 0 , respectively.As shown in Figure 5a, an inclusion was inserted near the weld toe line to discuss the stress distribution around the crack.The variable d shown in Figure 5b represents the relative position of the inclusion to the crack.The case where the crack passes along the center of the inclusion, i.e., d = 0, is discussed first.Figure 6 presents the stress distribution of cracks in two scenarios, without inclusion and with inclusions.The inclusions are differentiated by their elasticity modulus.The numerical results of the simulation indicate that the presence of particle inclusions has a significant impact on the stress state in the vicinity.The stress field of the crack is uniformly distributed along the center of the crack in a homogeneous material, as can be observed from Figure 6a.The phenomenon of asymmetry in stress fields around cracks caused by inclusions is directly linked to the elasticity modulus of inclusions.Figure 6b,c indicates that stress concentration primarily occurs inside the stiff inclusion, resulting in higher internal stress as compared to the basic material region.In the case of soft inclusion, the stress value inside the inclusion is not significant, but stress concentration occurs in the region close to the crack.The findings suggest that as the elasticity modulus of the inclusions increases, the stress of the matrix material near the crack initially surpasses that of the inclusions and then decreases below it.In the following section, we will explore the crack growth characteristics in different stress fields.

Crack SIFs Considering Crack-Inclusion
Researchers [39] found that the primary cause of fatigue failure at key welded connections of OSDs was the mode I crack.Therefore, the planar crack growth characteristics are first explored in the section.The stress state of different paths is important in understanding the different stages of crack growth.Figure 7 shows the variations in SIFs of the initial crack at different paths within homogeneous material.Due to software limitations, the SIFs on the 0 and the 1 growth paths may be distorted.Therefore, the SIFs on the 0.1 and 0.9 paths are primarily analyzed to approximately represent the SIF value at the tip of the major axis of the crack.The numerical results of the simulation indicate that the presence of particle inclusions has a significant impact on the stress state in the vicinity.The stress field of the crack is uniformly distributed along the center of the crack in a homogeneous material, as can be observed from Figure 6a.The phenomenon of asymmetry in stress fields around cracks caused by inclusions is directly linked to the elasticity modulus of inclusions.Figure 6b,c indicates that stress concentration primarily occurs inside the stiff inclusion, resulting in higher internal stress as compared to the basic material region.In the case of soft inclusion, the stress value inside the inclusion is not significant, but stress concentration occurs in the region close to the crack.The findings suggest that as the elasticity modulus of the inclusions increases, the stress of the matrix material near the crack initially surpasses that of the inclusions and then decreases below it.In the following section, we will explore the crack growth characteristics in different stress fields.

Crack SIFs Considering Crack-Inclusion
Researchers [39] found that the primary cause of fatigue failure at key welded connections of OSDs was the mode I crack.Therefore, the planar crack growth characteristics are first explored in the section.The stress state of different paths is important in understanding the different stages of crack growth.Figure 7 shows the variations in SIFs of the initial crack at different paths within homogeneous material.Due to software limitations, the SIFs on the 0 and the 1 growth paths may be distorted.Therefore, the SIFs on the 0.1 and 0.9 paths are primarily analyzed to approximately represent the SIF value at the tip of the major axis of the crack.
It can be observed from Figure 7 that the SIFs at the crack tip are symmetrically distributed around the center and will grow more obvious in the major axis direction of the crack.As the depth of the fatigue crack exceeds approximately half of the deck thickness (i.e., 8 mm), the growth rate of the deepest point decreases, leading to a gradual flattening of the crack.For inclusion cases, the variation in the SIF influence factor λ (λ = K I with inclusion /K I without inclusion ) on the 0.9 growth path of crack are concerned, as shown in Figure 8.In this study, the term 'the proximal point' and 'the remote point' are used to describe the crack tip in the major axis direction near and far from the inclusion, respectively.crack growth characteristics in different stress fields.

Crack SIFs Considering Crack-Inclusion
Researchers [39] found that the primary cause of fatigue failure at key welded connections of OSDs was the mode I crack.Therefore, the planar crack growth characteristics are first explored in the section.The stress state of different paths is important in understanding the different stages of crack growth.Figure 7 shows the variations in SIFs of the initial crack at different paths within homogeneous material.Due to software limitations, the SIFs on the 0 and the 1 growth paths may be distorted.Therefore, the SIFs on the 0.1 and 0.9 paths are primarily analyzed to approximately represent the SIF value at the tip of the major axis of the crack.It can be observed from Figure 7 that the SIFs at the crack tip are symmetrically distributed around the center and will grow more obvious in the major axis direction of the crack.As the depth of the fatigue crack exceeds approximately half of the deck thickness (i.e., 8 mm), the growth rate of the deepest point decreases, leading to a gradual flattening of the crack.For inclusion cases, the variation in the SIF influence factor λ (λ = KI with inclusion/KI without inclusion) on the 0.9 growth path of crack are concerned, as shown in Figure 8.In this study, the term 'the proximal point' and 'the remote point' are used to describe the crack tip in the major axis direction near and far from the inclusion, respectively.It can be concluded from Figure 8 that a soft inclusion results in the amplification of λ whenever a crack approaches the inclusion (x < 5.2 mm).After a crack enters the soft inclusion (5.2 mm ≤ x ≤ 8.2 mm), the amplification effect will be alleviated.A stiff inclusion ahead of a crack produces shielding on the approaching crack (x < 5.2 mm).As the crack penetrates the stiff inclusion, λ increases.Amplification and shielding effects produced by stiff and soft inclusions disappear soon after the crack tip leaves an inclusion (see the region for x > 8.2 mm).
The impact of spacing d between the inclusion and the crack on the shielding and amplification of SIFs is also discussed, as shown in Figure 9.It can be concluded from Figure 8 that a soft inclusion results in the amplification of λ whenever a crack approaches the inclusion (x < 5.2 mm).After a crack enters the soft inclusion (5.2 mm ≤ x ≤ 8.2 mm), the amplification effect will be alleviated.A stiff inclusion ahead of a crack produces shielding on the approaching crack (x < 5.2 mm).As the crack penetrates the stiff inclusion, λ increases.Amplification and shielding effects produced by stiff and soft inclusions disappear soon after the crack tip leaves an inclusion (see the region for x > 8.2 mm).
The impact of spacing d between the inclusion and the crack on the shielding and amplification of SIFs is also discussed, as shown in Figure 9.
According to Figure 9, it can be inferred that the shielding and amplification effect of inclusions on cracks is significant when d is small.The interference effect of inclusions on crack growth gradually diminishes with the increase in d.According to Figure 9, it can be inferred that the shielding and amplification effect of inclusions on cracks is significant when d is small.The interference effect of inclusions on crack growth gradually diminishes with the increase in d.

Effects of Crack-Inclusion Interaction on the Crack Shape
The shape change of the crack is influenced by the growth rate of crack tips.The crack growth rate and fatigue life, as shown in Equations ( 8) and ( 9), depend on the following parameters.According to recommendations from the BS7910 [40], BS7608 [41] and the IIW [37], C = 5.21 × 10 −13 N•mm −3/2 , m = 3, and th K Δ = 63 N•mm −3/2 .The initial depth a0 of the crack is assumed to be 0.5 mm.The crack size limit in the depth direction af is assumed half the plate thickness, i.e., 8 mm.According to the above parameters, the fatigue life curve of the crack shown in Figure 10 can be obtained.

Effects of Crack-Inclusion Interaction on the Crack Shape
The shape change of the crack is influenced by the growth rate of crack tips.The crack growth rate and fatigue life, as shown in Equations ( 8) and ( 9), depend on the following parameters.According to recommendations from the BS7910 [40], BS7608 [41] and the IIW [37], C = 5.21 × 10 −13 N•mm −3/2 , m = 3, and ∆K th = 63 N•mm −3/2 .The initial depth a 0 of the crack is assumed to be 0.5 mm.The crack size limit in the depth direction a f is assumed half the plate thickness, i.e., 8 mm.According to the above parameters, the fatigue life curve of the crack shown in Figure 10 can be obtained.
The results shown in Figure 10 indicate that the presence of inclusions has an impact on the rate of crack growth.Soft inclusions were found to accelerate the crack growth, while stiff inclusions were observed to inhibit it, from the overall crack growth process.Specifically, the fatigue life of the crack is 178 × 10 4 , 164 × 10 4 , and 196 × 10 4 cycles under no inclusion, stiff inclusion, and soft inclusion conditions, respectively.The interference effect exhibits an asymmetrical growth phenomenon, wherein the influence is greatest at the proximal point of the crack and least at the remote point.The results shown in Figure 10 indicate that the presence of inclusions has an impact on the rate of crack growth.Soft inclusions were found to accelerate the crack growth, while stiff inclusions were observed to inhibit it, from the overall crack growth process.Specifically, the fatigue life of the crack is 178 × 10 4 , 164 × 10 4 , and 196 × 10 4 cycles under no inclusion, stiff inclusion, and soft inclusion conditions, respectively.The interference effect exhibits an asymmetrical growth phenomenon, wherein the influence is greatest at the proximal point of the crack and least at the remote point.
Varying growth rates at crack tips shown in Figure 10 result in different shapes of crack growth.The evolution of the crack shape is described using the ratio of c/a of the crack, as shown in Figure 11.As observed in Figure 11, the fundamental laws remain unchanged regardless of the existence of inclusions.The values of c/a, which were initially 2, exhibited a sharp decline to 1.6, followed by a subsequent upward trend.The difference is that cracks expand Varying growth rates at crack tips shown in Figure 10 result in different shapes of crack growth.The evolution of the crack shape is described using the ratio of c/a of the crack, as shown in Figure 11.The results shown in Figure 10 indicate that the presence of inclusions has an impact on the rate of crack growth.Soft inclusions were found to accelerate the crack growth, while stiff inclusions were observed to inhibit it, from the overall crack growth process.Specifically, the fatigue life of the crack is 178 × 10 4 , 164 × 10 4 , and 196 × 10 4 cycles under no inclusion, stiff inclusion, and soft inclusion conditions, respectively.The interference effect exhibits an asymmetrical growth phenomenon, wherein the influence is greatest at the proximal point of the crack and least at the remote point.
Varying growth rates at crack tips shown in Figure 10 result in different shapes of crack growth.The evolution of the crack shape is described using the ratio of c/a of the crack, as shown in Figure 11.As observed in Figure 11, the fundamental laws remain unchanged regardless of the existence of inclusions.The values of c/a, which were initially 2, exhibited a sharp decline to 1.6, followed by a subsequent upward trend.The difference is that cracks expand As observed in Figure 11, the fundamental laws remain unchanged regardless of the existence of inclusions.The values of c/a, which were initially 2, exhibited a sharp decline to 1.6, followed by a subsequent upward trend.The difference is that cracks expand symmetrically about the center and reach a final c/a of 2.8 in a homogeneous material.Soft inclusions have minimal contribution to the final value of c/a, with a slight increase in c/a in area A during the growth process.However, stiff inclusions significantly impact the crack shape, with the value of c/a in area B being 22% larger than that of area A. The analysis result indicates that it is important to consider the impact of inclusions on the condition of the crack depth a from the measured surface crack length 2c.A schematic diagram of the evolution of the crack shape is shown in Figure 12.
symmetrically about the center and reach a final c/a of 2.8 in a homogeneous material.Soft inclusions have minimal contribution to the final value of c/a, with a slight increase in c/a in area A during the growth process.However, stiff inclusions significantly impact the crack shape, with the value of c/a in area B being 22% larger than that of area A. The analysis result indicates that it is important to consider the impact of inclusions on the condition of the crack depth a from the measured surface crack length 2c.A schematic diagram of the evolution of the crack shape is shown in Figure 12.It can be observed in Figure 12 that regardless of the presence of inclusions, cracks tend to develop and flatten out.These findings are consistent with the experimental law that indicates fatigue cracks have likely propagated to a significant length in the major axis direction once they penetrate the deck.During crack growth, inclusions can cause shape asymmetry.Stiff inclusions have a stronger impact on the crack shape, resulting in a flatter area away from the inclusions.

Evolution of Crack SIFs during Propagation
Representative fatigue cracks at DTR weld joints are mixed-mode cracks of mode Ⅰ, Ⅱ, and Ⅲ leading by Model Ⅰ [42].The inhomogeneity of the mechanical properties of the material caused by inclusions strengthens the attribute of mixed-mode cracks.Therefore, a 3D crack growth simulation was conducted to provide a more comprehensive exploration of the crack growth characteristics.For inclusion cases, a spherical inclusion with a radius of 1.5 mm was inserted on the deck plate near the weld toe line, i.e., d = r.
The SIF curves for the crack in homogeneous materials were initially studied, as shown in Figure 13.The results indicate that the fatigue cracks that initiate from the weld toe of RTD joints are mixed-mode cracks of mode I and mode III, and the cracks of mode-Ⅱ are negligible.It can be observed in Figure 12 that regardless of the presence of inclusions, cracks tend to develop and flatten out.These findings are consistent with the experimental law that indicates fatigue cracks have likely propagated to a significant length in the major axis direction once they penetrate the deck.During crack growth, inclusions can cause shape asymmetry.Stiff inclusions have a stronger impact on the crack shape, resulting in a flatter area away from the inclusions.

Evolution of Crack SIFs during Propagation
Representative fatigue cracks at DTR weld joints are mixed-mode cracks of mode I, II, and III leading by Model I [42].The inhomogeneity of the mechanical properties of the material caused by inclusions strengthens the attribute of mixed-mode cracks.Therefore, a 3D crack growth simulation was conducted to provide a more comprehensive exploration of the crack growth characteristics.For inclusion cases, a spherical inclusion with a radius of 1.5 mm was inserted on the deck plate near the weld toe line, i.e., d = r.
The SIF curves for the crack in homogeneous materials were initially studied, as shown in Figure 13.The results indicate that the fatigue cracks that initiate from the weld toe of RTD joints are mixed-mode cracks of mode I and mode III, and the cracks of mode-II are negligible.
To quantify the effect of SIFs for mode III on crack propagation, the variations of K III /K I were calculated under inclusions with different elasticity moduli, as shown in Figure 14.
It can be inferred from Figure 14 that values of K III /K I at the proximal and remote points exhibit different variation characteristics.The K III /K I at the proximal point shows an overall increasing trend due to the presence of stiff inclusions, as shown in Figure 14a.It is evident that the stiff inclusion results in a transient and noticeable amplification in the K III /K I value, and the magnitude of the amplification is directly proportional to the ratio of E 1 /E 0 .However, the soft inclusions enhance the K III /K I value more significantly at the remote point compared to the stiff inclusions as shown in Figure 14b.

SIF ratio variations
Crack length(mm)

SIF ratio variations
Crack length(mm) It can be inferred from Figure 14 that values of KIII/KI at the proximal and remote points exhibit different variation characteristics.The KIII/KI at the proximal point shows an overall increasing trend due to the presence of stiff inclusions, as shown in Figure 14a.It is evident that the stiff inclusion results in a transient and noticeable amplification in the KIII/KI value, and the magnitude of the amplification is directly proportional to the ratio of E1/E0.However, the soft inclusions enhance the KIII/KI value more significantly at the remote point compared to the stiff inclusions as shown in Figure 14b.

Effects of Crack-Inclusion on the Crack Growth Direction
The main crack in the joint weld generally grows perpendicular to the direction of stress loading.However, factors such as stress state and welding defects can impact the crack growth direction.As a result, the crack may deviate from the main growth direction and exhibit deflection, such as bifurcation and bridging [43].With the increase in the SIFs for mode III, the growth direction of cracks is affected by inclusions to varying degrees,

Effects of Crack-Inclusion on the Crack Growth Direction
The main crack in the joint weld generally grows perpendicular to the direction of stress loading.However, factors such as stress state and welding defects can impact the crack growth direction.As a result, the crack may deviate from the main growth direction and exhibit deflection, such as bifurcation and bridging [43].With the increase in the SIFs for mode III, the growth direction of cracks is affected by inclusions to varying degrees, as shown in Figure 15.
The main crack in the joint weld generally grows perpendicular to the direction of stress loading.However, factors such as stress state and welding defects can impact the crack growth direction.As a result, the crack may deviate from the main growth direction and exhibit deflection, such as bifurcation and bridging [43].With the increase in the SIFs for mode III, the growth direction of cracks is affected by inclusions to varying degrees, as shown in Figure 15.It can be seen from Figure 15 that the elasticity modulus of inclusions has a significant influence on the crack growth direction.The fatigue crack grows in a mixed mode and It can be seen from Figure 15 that the elasticity modulus of inclusions has a significant influence on the crack growth direction.The fatigue crack grows in a mixed mode and deviates from its original growth direction to varying degrees.For cases with no inclusion and stiff inclusion, the crack tends to be biased towards the welded area.However, stiff inclusion will result in a greater deflection compared to the case without inclusions.On the other hand, soft inclusion attracts the proximal point to grow towards the roof area, while repeling the remote point growing towards the welded area.It is important to note that the repulsion effect of soft inclusions at the remote point of the crack is more pronounced than that of stiff inclusions, which is consistent with the result calculated in Figure 14b.The above conclusions can be applied to control the direction of crack growth, effectively diverting it away from the most critical path and thereby delaying or preventing the occurrence of complete fracture to a significant degree.
The deflection of the crack growth direction is often accompanied by a decrease in the crack growth rate.As shown in Figure 16, the deflection of the crack growth direction has a more significant impact on the growth rate in the length direction compared to the crack depth direction.Specifically, the growth rate of the proximal point increases in the order of soft inclusion, no inclusion, and stiff inclusion.The growth rate of the remote point increases in the order of the soft inclusion, the stiff inclusion, and no inclusion.The phenomena demonstrate that individual inclusions have minimal impact on crack fatigue life.In addition, the degree of crack deflection resulting from inclusions has a negative correlation with the crack growth rate.deviates from its original growth direction to varying degrees.For cases with no inclusion and stiff inclusion, the crack tends to be biased towards the welded area.However, stiff inclusion will result in a greater deflection compared to the case without inclusions.On the other hand, soft inclusion attracts the proximal point to grow towards the roof area, while repeling the remote point growing towards the welded area.It is important to note that the repulsion effect of soft inclusions at the remote point of the crack is more pronounced than that of stiff inclusions, which is consistent with the result calculated in Figure 14b.The above conclusions can be applied to control the direction of crack growth, effectively diverting it away from the most critical path and thereby delaying or preventing the occurrence of complete fracture to a significant degree.The deflection of the crack growth direction is often accompanied by a decrease in the crack growth rate.As shown in Figure 16, the deflection of the crack growth direction has a more significant impact on the growth rate in the length direction compared to the crack depth direction.Specifically, the growth rate of the proximal point increases in the order of soft inclusion, no inclusion, and stiff inclusion.The growth rate of the remote point increases in the order of the soft inclusion, the stiff inclusion, and no inclusion.The phenomena demonstrate that individual inclusions have minimal impact on crack fatigue life.In addition, the degree of crack deflection resulting from inclusions has a negative correlation with the crack growth rate.

Conclusions
This study investigated crack-inclusion interaction for rib-to-deck welded joints in OSDs based on numerical simulation and linear elastic fracture mechanics.A refined finite element model of a half U-rib with cracks and inclusions was established by using the FRANC3D-ABAQUS interactive technology.The full processes of the crack-inclusion interaction from approaching and penetrating were accurately simulated.The effects of inclusion on significant parameters, such as the SIFs, the shape factor, the growth rate, and

Conclusions
This study investigated crack-inclusion interaction for rib-to-deck welded joints in OSDs based on numerical simulation and linear elastic fracture mechanics.A refined finite element model of a half U-rib with cracks and inclusions was established by using the FRANC3D-ABAQUS interactive technology.The full processes of the crack-inclusion interaction from approaching and penetrating were accurately simulated.The effects of inclusion on significant parameters, such as the SIFs, the shape factor, the growth rate, and the growth direction were analyzed.
The numerical results indicate that fatigue crack growth in welds is unstable due to inclusions.The stiff and soft inclusions amplify and shield the SIF of cracks when the crack grows to the local area of inclusions.During the entire process of crack growth, the soft and stiff inclusion accelerate and inhibit the crack growth, respectively.The magnitude of shielding and amplification effects is related E 1 /E 0 and the distance between the cracks and inclusions.Stiff inclusions lead to asymmetric growth of the crack shape.The flattening of the crack increases by 22% in regions farther away from the inclusion compared to those closer to it.Inclusions strengthen the attribute of mixed-mode cracks.Compared with the fatigue life, the deflection effect of inclusions on the crack growth direction deserves more investigation.Soft and stiff inclusions attract and repel the growth direction of the proximal point of the crack, respectively.Further investigations will be conducted in the following respects.The stochastic parameters of inclusion, such as the location, the size, and the shape are crucial for understanding crack growth characteristics.Experimental studies are necessary to validate the fatigue crack growth characteristics under welding defects.In the context of real bridge detection, it is crucial to consider the probabilistic characteristics of welding defects.The chemical composition of the steel, the welding method, and the microstructure of the material will be considered and investigated in future studies.

Figure 1 .
Figure 1.A finite width plate with semi-elliptical crack.

Figure 2 .
Figure 2. Dimension of a suspension bridge: (a) overall view; (b) a half cross-section.

Figure 2 .
Figure 2. Dimension of a suspension bridge: (a) overall view; (b) a half cross-section.

Figure 2 .
Figure 2. Dimension of a suspension bridge: (a) overall view; (b) a half cross-section.

Figure 4 .
Figure 4.The FE model developed by F-A technology.

Figure 4 .
Figure 4.The FE model developed by F-A technology.

Figure 8 .
Figure 8. Variation in the SIF influence factor λ on the 0.9 growth path.

Figure 8 .
Figure 8. Variation in the SIF influence factor λ on the 0.9 growth path.

Figure 10 .
Figure 10.Growth rate of crack of mode-Ⅰ.

Figure 11 .
Figure 11.The ratio of c/a as a function of a.

Figure 10 .
Figure 10.Growth rate of crack of mode-I.

Figure 10 .
Figure 10.Growth rate of crack of mode-Ⅰ.

Figure 11 .
Figure 11.The ratio of c/a as a function of a.

Figure 11 .
Figure 11.The ratio of c/a as a function of a.

Figure 12 .
Figure 12.Evolution of crack shape: (a) without inclusion; (b) with a soft inclusion; (c) with a stiff inclusion.

Figure 12 .
Figure 12.Evolution of crack shape: (a) without inclusion; (b) with a soft inclusion; (c) with a stiff inclusion.

Figure 13 .
Figure 13.SIFs of mixed-mode cracks in homogeneous materials (at the crack tip along the long axis of the crack).To quantify the effect of SIFs for mode III on crack propagation, the variations of KⅢ/KⅠ were calculated under inclusions with different elasticity moduli, as shown in Figure 14.

Figure 13 . 18 Figure 13 .
Figure 13.SIFs of mixed-mode cracks in homogeneous materials (at the crack tip along the long axis of the crack).

Figure 14 .
Figure 14.Variations in KIII/KI: (a) at the proximal point; (b) at the remote point.

Figure 14 .
Figure 14.Variations in K III /K I : (a) at the proximal point; (b) at the remote point.

Figure 15 .
Figure 15.The effect of inclusion on crack growth direction.

Figure 15 .
Figure 15.The effect of inclusion on crack growth direction.

Figure 16 .
Figure 16.Growth rate of mixed-mode cracks.

Figure 16 .
Figure 16.Growth rate of mixed-mode cracks.

Table 1 .
Comparison results of the plate example.

Table 1 .
Comparison results of the plate example.