Double cracks with single-inclusion fatigue propagation of surface-quenched large modulus rack

Abstract

A connectivity criterion based on the analysis of the stress intensity factor for the double cracks and inclusion was developed to study the residual life of large modulus crack. Applying the moving load, the finite element model in FRANC3D was proposed to study the variation law of the spacing ratio $R_{S}$ and stress intensity factor amplification coefficient $R_{K}$ . Also, the spacing ratio threshold and the initial spacing threshold for the law of propagation of the double cracks and inclusion were obtained. After the analysis of the threshold, the initial equivalent single crack $2 C_{E}^{1}$ was obtained based on the function $R_{K} (R_{S})$ and stress intensity factor for circular mode I crack $K_{I} (a)$ . The final equivalent single crack $2 C_{E}^{2}$ was obtained by innovative use of the fatigue load cycle correction coefficient $f (R_{S})$ . Moreover, the residual fatigue life of surface-quenched large modulus rack is analyzed to be 55 years. It is revealed that the design of the large modulus crack is safe for the normal operation of the Three Gorges Shiplift.

Keywords

Large modulus rack crack inclusion threshold connectivity criterion residual fatigue life

Introduction

The Three Gorges vertical shiplift adopts rack-and-pinion climbing type. Its driving gear and rack modulus is up to 62.667 mm. The rack is G35CrNiMo6-6. After quenching and tempering, the surface of the rack is quenched by induction quenching. On one hand, due to the scale, there are few engineering application cases, mature design theory, and norms in terms of the large modulus rack. The surface cracks and inclusion inevitably appear after rack machining. On the other hand, the safety factor methods of contact strength and bending strength of tooth roots are still applied to check the design of rack in engineering, and the Miner fatigue cumulative damage theory is used to estimate the fatigue life of rack. This method fails to guarantee the accurate results. Therefore, it is essential to study the fatigue crack growth of large modulus rack.

In recent years, the fatigue crack propagation and fatigue life in medium and small gear have been studied with various methods, including contact stress method, power density method, and the establishment of particle filter model.^1–3 For multiple crack propagation, the double crack growth is studied by extended finite element method.^4,5 Also, the interference effect of the double crack was investigated experimentally.⁶ Based on the peridynamics theory, an analysis of the collinear double cracks containing the initial defect was conducted.⁷ For the coalescing of the multiple cracks, the experimental and numerical investigations are utilized in studying the semielliptical crack propagation.^8–10

Regarding the research on fatigue crack propagation characteristics of gear/crack heat treatment, the effect of quenching on crack growth rate of metal materials was investigated.^11–13 The residual stress distribution on the crack propagation of the gear root was analyzed on the basis of the electromagnetic-thermal-metallic-mechanical coupled numerical model.¹⁴ In addition, the variations of hardness and residual stress introduced by carburizing and quenching processes have remarkable effects on the rolling contact fatigue for gear meshing.¹⁵

However, the researches of the gear/rack crack propagation involved in the large modulus, as well as the initial cracks and inclusions, have not been found. Besides, few researches have been conducted about the influence of heat treatment on multiple cracks/inclusions propagation.

By applying moving load on the model in FRANC3D and studying the possible crack distribution rules on the large modulus rack, the interference effects between the double cracks and inclusion were analyzed quantitatively. Subsequently, a connectivity criterion was proposed to prospect the residual fatigue life of the large modulus rack.

Threshold of double cracks propagation with single inclusion

Cracks propagation model

To precisely simulate the mesh of gear/crack in working condition, a moving load method is introduced. In Figure 1, the rack is meshed with the pinion of the drive pinion bracket on the shiplift. Figure 1 signifies that each tooth surface of the load is discretized into 15 loading regions, consisting of 6 double teeth-meshing regions, 3 single teeth-meshing regions, and 6 double teeth-meshing regions, arranging from the top to the root of the tooth. The load of 15 loading regions of the rack can be computed by the gear meshing principle.¹⁶ The boundary size of the rack model is the multiple of rack modulus. The width of rack is 800 mm, and the tooth height is 141 mm. The fatigue crack propagation analysis processes in FRANC3D are as follows:

The finite element simulation model for the large modulus crack containing the initial double cracks and the inclusion is proposed (Figure 1).

The stress of the finite simulation model for the large modulus crack containing the initial double cracks and the inclusion is calculated by finite element theory.

Reading the stress analysis results, compute stress intensity factor (SIF) of the front points of the cracks, analyze crack propagation, update crack front position, and re-mesh the finite element model in FRANC3D.

Repeat steps 1–3, until the end of the fatigue crack propagation.

Figure 1.

Simulation model of large modulus crack.

According to the results of crack and inclusion detection of large modulus rack, it is assumed that the inclusion on the surface is a hemispherical cavity with a diameter of $2 R_{0} = 6 mm$ approximately and located at the middle of the rack root where the cracks distribute primarily; the surface parallel collinear cracks are symmetrically arranged on both sides of the inclusion, with the initial length $2 C_{0} = 6 mm$ , and the initial depth $a_{0} = 3 mm$ (Figure 1).^17,18 The real shape of the inclusions and cracks during the processing and detecting are shown in Figure 2. The distance between the right end of the left crack (or the left end of the right crack) and the left end of the inclusion (or the right end of the inclusion) is $S$ , and the crack spacing ratio is $R_{S} = S / (2 C)$ . $2 C$ is the changing length of the crack in the crack propagation. According to the statistical results based on shape and distribution law of large modulus rack crack, double cracks and inclusion in the cases of the horizontal spacing $S_{0} = 2 C_{0}$ , $8 C_{0}$ , $14 C_{0}$ , $20 C_{0}$ , and $26 C_{0}$ are simulated.

Figure 2.

Defects of large modulus rack: (a) inclusions and (b) cracks.

Spacing ratio threshold

Table 1 signifies the variable law of SIF in the fatigue crack propagation of large modulus crack with the initial spacing between inclusion and double cracks $S_{0} = 2 C_{0}$ , $8 C_{0}$ , $14 C_{0}$ , $20 C_{0}$ , and $26 C_{0}$ .

Table 1.

Variable law of SIF in the fatigue crack propagation of large modulus crack.

$S_{0}$	Propagation step	SIF of various points (MPa $\sqrt{mm}$ )
		K	J	M	N
$2 C_{0}$	First step	85.6	86.8	86.9	86.2
	Fifth step	98	105.7	103	97.4
	Tenth step	93.8	112.3	105.6	83.5
$8 C_{0}$	First step	85.6	85.3	85.5	85.8
	Fifth step	94.7	95.8	97.6	96.2
	Tenth step	97.9	99.7	99.4	99.1
$14 C_{0}$	First step	85.5	85.8	85.7	85.5
	Fifth step	92.1	93.4	92.1	91.5
	Tenth step	98.2	98.4	97.7	97.3
$20 C_{0}$	First step	85.2	85.7	85.5	85.2
	Fifth step	94.5	94.8	94.6	94.5
	Tenth step	97.5	97.9	97.8	97.6
$26 C_{0}$	First step	85.1	85.4	85.6	84.8
	Fifth step	92.7	93.2	93.8	92.7
	Tenth step	97.3	97.8	98.1	97.7

SIF: stress intensity factor.

As shown in Table 1, the SIF at the left and right ends of the initial crack are equal at the beginning of propagation, indicating no interference in the propagations of the double cracks and inclusion. With the increase in the propagation step, the SIF of the right end of the left crack is greater than that of the left crack, and the SIF of the left end of the right crack is greater than that of the right crack. This indicates that more intense propagation begins to occur in the ends of double cracks near inclusion. Thus, the increasing propagation step results in the smaller initial spacing between double cracks and inclusion, and the greater difference of SIF for the two ends of the same crack (left crack or right crack of inclusion).

For the assessment on the SIF reinforcement effect, it is useful to determine the SIF amplification coefficient $R_{K} = K_{TRI} / K_{SIN}$ . $K_{TRI}$ is the SIF for the ends of the double cracks near inclusion, and $K_{SIN}$ is the SIF for a single crack propagating at the same initial position with the same propagation steps.

$R_{K}$ and $R_{S}$ are calculated in various initial spacing $S_{0} = 2 C_{0}$ , $8 C_{0}$ , $14 C_{0}$ , $20 C_{0}$ , and $26 C_{0}$ , respectively, and the variable law of $R_{K}$ and $R_{S}$ is obtained in the propagation (Figure 3). From the same five curves in Figure 3, it is believed that the enforcement effect of $R_{K}$ is basically unconcerned with $S_{0}$ and only related to $R_{S}$ . The functional relationship between $R_{K}$ and $R_{S}$ is fitted by the use of the nonlinear least squares method in the form

R_{K} = 1.0359 R_{S}^{- 0.02021}

(1)

Figure 3.

Different initial spacing: relationship between $R_{K}$ and $R_{S}$ .

On the basis of Figure 3 and variation of derivative function of equation (1), some critical points can be seen: when $R_{S} > 5.7$ , $R_{K} \approx 1$ , there are no interference between double cracks and inclusion and the three cracks are isolated; when $1.5 < R_{S} < 5.7$ , $1 < R_{K} < 1.028$ , the interference begins to appear in the double cracks and inclusion, in spite of its little degree; when $0 < R_{S} < 1.5$ , $R_{K} > 1.028$ , the interference becomes severe and the reinforcement effect becomes remarkable. Therefore, 1.5 is the ratio threshold of spacing between double cracks and inclusion. When the double cracks and inclusion propagates to this threshold, they begin to interact.

Initial spacing threshold

Fatigue load cycle number of large modulus rack is calculated when the double cracks and inclusion propagate to the spacing ratio threshold, with the various initial spacing $S_{0} = 8 C_{0}$ , $14 C_{0}$ , $20 C_{0}$ , and $26 C_{0}$ . In this section, initial spacing $S_{0} = 2 C_{0}$ is not considered because the spacing ratio at this point is lower than 1.5, and interference is extremely intense. The calculation formula of the number of fatigue load cycles can be written in the form¹⁹

\frac{d a}{d N} = Ca \sqrt{\frac{1}{2} (\frac{Δ K^{2} (a) - Δ K_{th}^{2} (a)}{Δ K_{th}^{2} (a)} + \frac{K_{C}^{2} (a) - Δ K^{2} (a)}{K_{C}^{2} (a)})}

(2)

where $Δ K_{th}$ is the crack growth threshold and $Δ K_{C}$ is the fracture toughness. These two parameters are associated with the surface hardness of the large modulus rack. Based on the defect detection of the large modulus rack, surface hardness function $H$ from rack face to core can be written as²⁰

H = 610 - 20.3 a, (a \leq 17.6)

(3)

where $a$ is the depth of the crack propagation. The fracture toughness of the large modulus rack after quenching is arranged in the form²¹

β = \frac{1}{h} \times \ln (H_{1} / H_{0})

(4)

K_{IC} (a) = K_{IC 0} e^{β \cdot (h - a)}, (3 \leq a \leq 17.6)

(5)

where $H_{1}$ is the hardness of the core for the large modulus rack, $H_{0}$ is the surface hardness of the large modulus rack, h is the hardness change thickness of the large modulus rack, and $K_{IC 0} = 2620 MPa \sqrt{mm}$ is the fracture toughness of core (base material) of large modulus rack casting. From equation (3), the rack surface hardness is $610 HV$ , and the hardness of rack core is $252 HV$ (in Figure 4).

Figure 4.

Rack hardness distribution.

The crack growth threshold of the large modulus rack after quenching is arranged in the form²²

Δ K_{th} (a) = 3.4 \times 10^{- 3} (730 - 20.3 a) a^{\frac{1}{3}}, (3 \leq a \leq 17.6)

(6)

Next, the application of the integral operator for equation (2) leads to the following equation

\int_{a_{0}}^{a} \frac{1}{Ca \sqrt{\frac{1}{2} (\frac{Δ K^{2} (a) - Δ K_{th}^{2} (a)}{Δ K_{th}^{2} (a)} + \frac{K_{C}^{2} (a) - Δ K^{2} (a)}{K_{C}^{2} (a)})}} d a = \int_{0}^{N} d N = N

(7)

Due to the complex structure of integral formula, the solution of equation (7) is completed by the compound trapezoid formula.

In Figure 5, the variable law between the number of fatigue load cycles $N$ and $R_{S}$ can be obtained by the comprehensive use of equations (2)–(7). When $S_{0} = 8 C_{0}$ , $R_{S}$ changes from 4 to 1.5, and the fatigue load cycle number $N = 588, 240$ ; when $S_{0} = 14 C_{0}$ , $R_{S}$ changes from 7 to 1.5, and the number of fatigue load cycle $N = 657, 700$ ; when $S_{0} = 20 C_{0}$ , $R_{S}$ changes from 10 to 1.5, and $N = 781, 110$ ; when $S_{0} = 26 C_{0}$ , $R_{S}$ changes from 13 to 1.5, and $N = 807, 190$ .

Figure 5.

Fatigue load cycle number of double cracks and inclusion.

The design life of the Three Gorges shiplift is $70$ years, with working 335 days per year and operating 18 times (each time contains a rise and a fall) per day on average. Thus, its total fatigue load cycle: $N = 70 \times 335 \times 18 \times 2 = 844, 200$ . If the spacing ratio between double cracks and inclusion $R_{S}$ does not reach 1.5 within the design life of the shiplift, it is believed that no interference exists in the double cracks and inclusion. In the case of the maximum initial spacing, $S_{0} = 26 C_{0}$ , $N = 807, 190 < N = 844, 200$ . However, it is still convinced that the safest initial spacing threshold is $S_{0} = 26 C_{0}$ .

Connectivity criterion of double cracks with single inclusion

Initial equivalent crack length

For circular mode I crack on the surface of a finite body, the general expression of SIF is given by²³

K_{I} = [1 + 0.12 {(1 - \frac{a}{2 C})}^{2}] \frac{σ \sqrt{π a}}{Φ}

(8)

where $σ$ is the nominal stress, and $Φ$ is the second kind of elliptic integral, in the form

Φ = \int_{0}^{π / 2} {(1 - \frac{C^{2} - a^{2}}{C^{2}} si n^{2} φ)}^{1 / 2} d φ

(9)

where $φ$ varies from $0$ to $π / 2$ (in Figure 6).

Figure 6.

Semicircular crack.

Overall, the series expansion of $Φ$ can be written as

Φ \approx \frac{π}{2} [1 - \frac{1}{4} \frac{C^{2} - a^{2}}{C^{2}} - \frac{3}{64} {(\frac{C^{2} - a^{2}}{C^{2}})}^{2}]

(10)

Based on the results in various initial spacing, the relationship of $a$ and $2 C$ is fitted by non-linear least squares method, in the form

2 C = 0.886 a^{1.1894}

(11)

From equation (11), $(a / C) << 1$ , that is, the third item in equation (10) accounts for only about 5%. If omitted, the following can be obtained

Φ = \frac{3}{8} π + \frac{π}{8} \frac{a^{2}}{C^{2}}

(12)

On the basis of equations (8)–(12), the $K (a)$ is fitted to exponential function, in the form

K (a) = K_{SIN} = σ \sqrt{π} 0.766 a^{0.5749}

(13)

Now, equivalent initial crack length after fusion of double cracks and inclusion is obtained according to $R_{K}$ , assuming $K_{TRI}$ as SIF of single crack after equating, in the form

K_{TRI} = (R_{K}) K_{SIN} = σ \sqrt{π} 0.766 {(a_{E}^{1})}^{0.5749}

(14)

According to equations (13) and (14), the depth of a single crack after equating based on the amplification effect of SIF is defined as $a_{E}^{1}$ , in the form

a_{E}^{1} = {R_{K}}^{1 / 0.5749} a

(15)

Therefore, the length of a single crack after equating based on the amplification effect of SIF is defined as $2 C_{E}^{1}$ , in the form

2 C_{E}^{1} = 0.886 a^{1.1894} \times {R_{K}}^{4.1562}

(16)

Fatigue load cycle correction coefficient

The double cracks and inclusion will eventually fuse into a single crack with length of $2 C_{HB}$ (Figure 7(a)). For a single crack after equating based on the amplification effect of SIF, the number of fatigue load cycles required for a propagation from $2 C_{E}^{1}$ to $2 C_{HB}$ is defined as $N_{E}^{1}$ (Figure 7(b)). For a single crack length $2 C$ in the middle of tooth root, the number of fatigue load cycles is defined as $N_{SIN} (2 C)$ . Based on the results derived from equations (7) and (11), for a single crack length $2 C$ in the middle of tooth root, $N_{SIN} (2 C)$ can be fitted by the nonlinear least square method, in the form

\begin{matrix} N_{SIN} (2 C) = & - 0.0032 {(2 C)}^{4} + 1.5782 {(2 C)}^{3} \\ - 280.2302 {(2 C)}^{2} + 26, 139 (2 C) - 73, 096 \end{matrix}

(17)

Figure 7.

Combination and propagation of double cracks and inclusion: (a) double cracks and inclusion merge into single crack $2 C_{HB}$ after $N_{TRI}$ fatigue load cycles, (b) equivalent single crack $2 C_{E}^{1}$ propagates into single crack $2 C_{HB}$ after $N_{E}^{1}$ fatigue load cycles, and (c) equivalent single crack $2 C_{E}^{2}$ propagates into single crack $2 C_{HB}$ after $N_{E}^{2}$ fatigue load cycles.

$N_{E}^{1}$ can be defined as follows

N_{E}^{1} = N_{SIN} (2 C_{HB}) - N_{SIN} (2 C_{E}^{1})

(18)

The final equivalent crack length corrected by the fatigue load cycle correction coefficient is defined as $2 C_{E}^{2}$ , which can be written as

2 C_{E}^{2} = f (R_{S}) 2 C_{E}^{1}

(19)

where $f (R_{S})$ is the correction coefficient of fatigue load cyclic number.

Next, the number of fatigue load cycles required for a propagation from $2 C_{E}^{2}$ to $2 C_{HB}$ is defined as $N_{E}^{2}$ (Figure 7(c)), in the form

N_{E}^{2} = N_{SIN} (2 C_{HB}) - N_{SIN} (2 C_{E}^{2})

(20)

For double cracks and inclusion, the number of fatigue load cycles required for a single crack with a final fusion growth of $2 C_{HB}$ is defined as $N_{TRI}$ . In accordance with the crack equivalence criterion, $N_{E}^{2}$ is supposed to be equivalent with $N_{TRI}$ , which can be written as

N_{E}^{2} = N_{TRI}

(21)

Finally, the other form of $N_{SIN} (2 C_{E}^{2})$ is given by

N_{SIN} (2 C_{E}^{2}) = N_{E}^{1} - N_{TRI} + N_{SIN} (2 C_{E}^{1})

(22)

If $N_{E}^{1}$ , $N_{TRI}$ , and $2 C_{E}^{1}$ are determined, there will be only one unknown coefficient $f (R_{S})$ . For each $2 C_{E}^{1}$ , there will be a definite $f (R_{S})$ corresponding to it (Table 2). Thus, $f (R_{S})$ can be fitted, in the form

f (R_{S}) = 0.0126 R_{S}^{3} - 0.2537 R_{S}^{2} + 3.6778 R_{S} + 1.0221

(23)

When $S_{0} = 26 C_{0}$ , $2 C_{HB} = 183.50 mm$ , close to a quarter of the length of the large modulus rack. Thus, considering the operating condition is extremely dangerous, this initial spacing is not discussed in Table 2.

Table 2.

Correction coefficient of fatigue load cyclic number for $S_{0} = 2 C_{0}$ , $8 C_{0}$ , $14 C_{0}$ , and $20 C_{0}$ .

$S_{0}$	$2 C_{HB}$	$R_{S}$	$R_{K}$	$a$	$2 C_{E}^{1}$	$N_{TRI}$	$N_{E}^{1}$	$f (R_{s})$
$2 C_{0}$	$50.15$	$1.002$	$1.014$	$2.997$	$12.928$	$287, 900$	$475, 190$	$1.6705$
		$0.373$	$1.047$	$3.462$	$18.247$	$180, 870$	$346, 460$	$1.7266$
		$0.166$	1.093	$3.710$	$21.527$	$107, 490$	$288, 960$	$1.9182$
		$0.068$	$1.111$	$3.840$	$23.374$	$76, 452$	$262, 490$	$1.9041$
$8 C_{0}$	$84.32$	$3.838$	$1.004$	$2.997$	$6.349$	$594, 250$	$2, 625, 900$	$12.345$
		$2.384$	$1.002$	$3.465$	$8.348$	$522, 780$	$2, 543, 400$	$9.4845$
		$1.832$	$1.000$	$3.811$	$9.966$	$472, 020$	$2, 483, 500$	$7.9998$
		$1.279$	$0.994$	$4.194$	$11.741$	$432, 290$	$2, 424, 400$	$6.8262$
$14 C_{0}$	$118.98$	$7.343$	$0.986$	$2.997$	$5.934$	$669, 630$	$9, 721, 100$	$19.681$
		$5.320$	$0.968$	$3.350$	$6.895$	$643, 820$	$9, 679, 200$	$16.951$
		$4.092$	$0.993$	$3.758$	$9.437$	$544, 180$	$9, 579, 200$	$12.420$
		$3.368$	$0.982$	$4.134$	$10.911$	$511, 400$	$9, 528, 000$	$10.752$
$20 C_{0}$	$152.93$	$10.011$	$0.996$	$2.997$	$6.086$	$700, 410$	$25, 154, 000$	$24.939$
		$6.485$	$1.004$	$3.554$	$8.697$	$596, 570$	$25, 046, 000$	$17.471$
		$5.453$	$0.999$	$3.858$	$10.010$	$560, 850$	$24, 998, 000$	$15.186$
		$4.600$	$1.024$	$4.217$	$12.970$	$481, 020$	$24, 904, 000$	$11.730$

Connectivity criterion

The calculating procedure of the connectivity criterion proposed in this article is shown as follows:

Based on the detection results for the large modulus rack, the spacing ratio between double cracks and inclusion $R_{S}$ is calculated. If $R_{S}$ is in the interference zone, the next step will be performed; otherwise, no effect can be found between the crack and the inclusion (for the large modulus rack of the Three Georges shiplift, $R_{S} \leq 1.5$ ).

The SIF amplification coefficient $R_{K}$ is calculated, and the length of a single crack after equating based on the amplification effect of SIF $2 C_{E}^{1}$ is obtained.

Based on equation (23), the fatigue load cycle correction coefficient $f (R_{S})$ is calculated. From equation (19), the final equivalent crack length corrected by the fatigue load cycle correction coefficient $2 C_{E}^{2}$ is obtained.

Based on the initial spacing of the double cracks and inclusion, the final equivalent crack depth corrected by the fatigue load cycle correction coefficient $a_{E}^{2}$ is calculated in accordance with equation (11).

Fatigue life of surface-quenched rack

Maximum allowable crack length

For a single crack length $2 C$ in the middle of tooth root, the propagation of the crack can be divided into the three stages on the basis of the slope change of equation (17), including low speed propagation $(2 C < 2 C_{1} = 30 mm)$ , middle speed propagation $(2 C_{1} < 2 C < 2 C_{2} = 143 mm)$ , and high speed propagation $(2 C > 143 mm)$ .

As shown in Figure 8, $2 C_{2} = 143 mm$ is the critical length for the rack failure. However, it should be noticed that some factors, including characterization error, stress distribution, and fracture toughness, are taken into account for judging whether the crack is failure or not. The maximum allowable crack length $2 C_{\max}$ is corrected by containing these factors, in the form

2 C_{\max} = \frac{2 C}{N_{BZ} \times N_{KI} \times N_{YL}}

(24)

where $N_{BZ}$ is defined as the safety coefficient for defect characterization, $N_{KI}$ is defined as the safety coefficient for the fracture toughness of material, and $N_{YL}$ is defined as the stress safety coefficient: $N_{BZ} = 1.1$ , $N_{KI} = 1.2,$ and $N_{YL} = 1.5$ .²⁰ Therefore, the maximum allowable crack length of the rack is $2 C_{\max} = 72.22 mm$ .

Figure 8.

Relationship between $N$ and $2 C$ for the propagation of the single crack.

Forecast of fatigue life

The procedure of the forecast of fatigue propagation life is shown as follows:

In accordance with the connectivity criterion (in 3.3), the double cracks and inclusion are combined into an equivalent single crack.

Based on the equivalent single crack length, the absolute safety of the large modulus rack is determined.

After comparison between the equivalent single crack length and the maximum allowable crack length $2 C_{\max}$ , the service life and the residual service life are determined.

Calculating the equivalent crack growth in the residual service life helps to judge whether the ultimate crack growth reaches the maximum allowable crack length in the residual service life or not.

For the Three Gorges Vertical shiplift servicing for $25$ years, the length of double cracks is $6 mm$ , the radius of the inclusion is $6 mm$ , and the spacing between the double cracks and the inclusion is $3 mm$ . In accordance with the connectivity criterion, the final equivalent crack length $2 C_{E}^{2} = 20.61 mm$ , with the time required for crack propagation to the maximum length being $59$ years. The forecast residual life of the Three Gorges Vertical shiplift is greater than its design residual service life of $55$ years.

However, the whole service life of the large modulus rack in the shiplift is also analyzed by Miner fatigue cumulative damage theory that is a conventional method. The equivalent load of the rack $F_{eq}$ can be written as

F_{eq} = {(\frac{N_{1} F_{1}^{p} + N_{2} F_{2}^{p}}{N_{1} + N_{2}})}^{1 / p}

(25)

where $p = 6.6$ is the experimental index of rack material; $F_{1}$ is the load of rack subjected to deviation water depth 0.1 m; $F_{2}$ is the load of rack subjected to deviation water depth 0.05 m; $N_{1}$ and $N_{2}$ are the loading cycle number of $F_{1}$ and $F_{2}$ , respectively; and $F_{eq} = 833.4 kN$ . Afterward, according to ISO 6336-6: 2006,²⁴ the safety coefficient of the bending fatigue strength on the rack root is $S_{F} = 3.50$ , and the safety coefficient of the contact fatigue strength on the rack root is $S_{H} = 1.57$ . Therefore, the whole service life of the large modulus rack $T_{1} = 70 \times 1.57 \approx 110$ years, which is more conservative than the results through using the method in this article ( $T_{2} = 25 + 55 = 80$ years).

Conclusion

The key observations are summarized as follows:

The variation law of the SIF amplification coefficient $R_{K}$ , changing with spacing ratio between the double cracks and inclusion $R_{S}$ , is obtained. If $0 < R_{S} < 1.5$ , $R_{K} > 1.028$ and the interference effect between the double cracks and the inclusion becomes obvious. If $1.5 < R_{S} < 5.7$ , $1 < R_{K} < 1.028$ , and the interference effect appears between double cracks and the inclusion. However, its strengthening effect is not obvious. If $R_{S} > 5.7$ , $R_{K} < 1$ , and almost no interference effects exist between the double cracks and the inclusion.

In the case that the initial crack size $a_{0} = 3 mm$ , $2 C_{0} = 6 mm$ , and the initial inclusion radius $R = 3 mm$ , if $S_{0} > 26 C_{0}$ , within $70$ years of the design life of the shiplift, $R_{S}$ will possibly not reach 1.5 with the increase of $S_{0}$ . In this case, the double cracks and the inclusion can be found isolated.

In order to obtain a single crack equivalent to the double cracks and the inclusion, the connectivity criterion is proposed. At first, the initial equivalent crack length is calculated with the SIF amplification coefficient. Subsequently, in accordance with the equivalent principle of the number of fatigue load cycles, the final equivalent crack length is obtained through the use of correction coefficient of fatigue load cyclic number $f (R_{S})$ .

For the Three Gorges Vertical shiplift, in the case that the initial crack size $a_{0} = 3 mm$ , $2 C_{0} = 6 mm$ , and the initial inclusion radius $R = 3 mm$ , the predicted residual life is $59$ years based on the proposed connectivity criterion, longer than its design life. Therefore, the length of cracks and inclusion meets the design requirement.

Footnotes

Handling Editor: James Baldwin

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The work presented in this article was supported by the National Key R & D Program of China (No. 2016YFC0402002), and the authors gratefully acknowledge this support.

ORCID iD

Xionghao Cheng

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