Recent Developments on the Unified Fatigue Life Prediction Method Based on Fracture Mechanics and its Applications

Abstract

Safety analysis and prediction of a marine structure is of great concern by many stakeholders and the general public. In order to accurately predict the structural reliability of an in-use marine structure, one needs to calculate accurately the fatigue crack growth at any service time. This can only be possible by using fracture mechanics approach and the core of fracture-mechanics-based method is to establish an accurate crack growth rate model which must include all the influential factors of the same order of sensitivity index. In 2011, based on the analysis of various influencing factors, the authors put forward a unified fatigue life prediction (UFLP) method for marine structures. In the following ten years of research, some further improvements of this method have been made and the applications of this UFLP are carried out. In this paper, these progresses are reported and its underlying principles are further elaborated. Some basic test data used to determine model parameters are also provided.

1. Introduction

Safety analysis and prediction of a marine structure is of great concern by many stakeholders including owners/investors, designers, manufacturers, service providers/operators, service users/cargo owners, regulatory organizations or charterers, and finally the general public [1]. As the technology progresses, the marine structure becomes larger and larger and its consequence for a failure becomes more and more severe. Researchers attach importance to reliability-based optimization for ships and offshore structures for safety [2].

In general, the safety of a marine structure can be expressed by the following mathematical model,

G = R − S > 0

(1)

where R is the resistance of the structure to the external loading, S is the external loading acted on the structure, and G is called the limit state. As experience tells us that the resistance of a marine structure is a function of time. It will decrease with the corrosion and fatigue development.

Due to the harsh marine environment and the fluctuating nature of wave loading, as well as the extensive use of welding technology and high-strength materials, various types of marine structures are easy to be plagued by fatigue cracks in the service process which are jointly caused by processing and manufacturing defects and the cyclic loading. The disadvantage of damage cumulative method which is applied in current ship classification rules is that the initial defects of materials and the final failure state are not defined clearly. The inherent theoretical defects make the prediction results very dispersive. A benchmark study has been made in literature on fatigue strength assessment procedures adopted by ship classification societies indicated a large difference in the predicted fatigue lives for a very simple detail [3]. In order to reasonably take account of the influence of fatigue cracks on the ultimate strength of a marine structure, accurate prediction of fatigue cracks is a necessity and this makes fracture mechanics an indispensable analytical tool, accordingly more and more researchers point out the problem of crack growth of offshore structures [2,3,4,5,6,7,8]. The core of fracture-mechanics-based method is to establish a crack growth rate model which must be able to explain a variety of fatigue phenomena, and the application of the model depends on the accurate grasp of the real environmental condition and loading sequence.

There have been a lot of researches on crack growth rate models since the Paris law being put forward [9]. The expression of Paris law is da/dN = C(∆K)ⁿ. In this expression, a is the crack depth or length; N is the number of stress cycles; C and n are parameters related to materials; and ∆K is stress intensity factor range. From then on, a new method to estimate the crack growth life is provided for fatigue research. Based on this, the damage tolerance design is developed, which makes the two disciplines of fracture mechanics and fatigue gradually combined. Although Paris law has been used widely in engineering because of its simple expression, it can only express the stable stage of crack growth, so it needs to simplify the analysis conditions in engineering application, and many later models have insufficient ability to explain various fatigue phenomena. In 2011, based on the quantitative analysis of various influencing factors [10,11,12,13], the authors put forward a unified fatigue life prediction method for marine structures [14]. In the following ten years of research, further improvements of this method on explaining more phenomenon have continued [15,16,17,18,19,20,21,22] and various applications of this method have been carried out [23,24,25,26,27,28,29,30]. Figure 1 gives a schematic presentation of the unified fatigue life prediction method. Firstly, the error sources in the process of fatigue life prediction are divided into four aspects: material (M), structure (S), environment (E), and load (L). When the model is used to calculate the fatigue life, the initial and final states of cracks should be clearly specified. These two states are closely related to the above four aspects. Then the unified crack growth rate model includes crack parameters, material strength, fracture parameters, load ratio, environmental parameters etc. In the follow-up study, the author’s research group has expanded the research on the influence of overload/underload effect [15,16], load sequence effect [17,18], small crack effect [19,20,21], creep effect [22], and other factors. With the improvement of the model, its ability to explain the phenomenon is stronger, and it can accurately predict the decay pattern of structural strength with crack growth, so as to be applied to the safety assessment of the structure and the formulation of maintenance and repair cycle. These methods have also been used to evaluate the crack growth of ship structures, offshore structures, and pressure hulls [17,18,31].

In this paper, the research works in recent years based on the unified fatigue life prediction method are summarized, and its underlying principles are further elaborated. Detailed introduction of extensive researches on application of the method in recent years are made as well as basic test data used to determine model parameters.

2. The Unified Fatigue Life Prediction Method and Its Further Developments

2.1. The General Procedure of UFLP

A relatively accurate calculation model of crack growth rate is the key issue to unified fatigue life prediction (UFLP), but before the life assessment of damaged structures with cracks, it is necessary to clarify the service environment of the structure, the future’s load history, the characteristics of the structure, and the basic properties of the material, which are input preparations for assessment procedure. Generally, it is necessary to carry out crack growth rate tests under different load ratios to obtain sufficient data, but this is very time consuming and expensive to be afforded. In the UFLP method, the relationship between different crack growth rate curves under different load ratios has been utilized. It is enough to obtain only one group of crack growth rate data, for example, under the condition of R = 0, including threshold value of stress intensity factor range. Other groups of crack growth rate curves under different load ratios can be well transformed. The determination of test scheme is also based on environmental and load conditions. To evaluate the long-term life of underwater structures, it is necessary to conduct testing in seawater to consider the effect of corrosion. The load history is particularly important. Marine structures always experience cyclic loading during their service period. Their fatigue crack growth rate behaviors have been proved from laboratory tests to be sensitive to the loading sequence encountered [17,18]. A standard load-time history (SLH) can be established to include typical features of the loading environment of a certain class of structures. The structure type and crack form together constitute the damaged structure characteristics. Furthermore, with the requirement of lightweight underwater structures such as manned cabin in a deep manned submersible, high-strength materials are widely used in marine structures [32,33,34,35]. The increase of strength usually leads to the decrease of fatigue life, and the crack is easier to produce. The crack growth rate of high strength steel is usually faster. All of these factors will be the basis considerations. Figure 2 provides the general procedure of UFLP with the considerations of the input preparation and the calculation method presented by Cui et al. [14]. The original crack growth rate model was derived based on McEvily model [36,37] and thus called the modified McEvily model.

2.2. The Original Modified McEvily Model

The initial expression of the model considers the variation of the unstable propagation condition with the crack length, the stress-strain constraint factor at the crack tip, and then the influence of residual stress, the effect of overload and underload [15,16], as well as the relationship between the crack growth threshold under different load ratios. A unified expression can be written in the following form,

$$\frac{\mathrm{d}a}{\mathrm{d}N}=\frac{A\cdot {\left(K_{\max }(1-\mathsf{\Phi }\cdot f_{\mathrm{op}})-\Delta K_{\mathrm{effth}}\right)}^m}{1-{\left(\raisebox{1ex}{$K_{\max }$}\!\left/ \!\raisebox{-1ex}{$K_{\mathrm{cf}}$}\right.\right)}^n}$$

(2)

where,

$$K_{\max }=\sqrt{\pi r_{\mathrm{e}}\left(Sec\frac{\pi }{2}\frac{{\sigma }_{\max }}{{\sigma }_{\mathrm{v}}}+1\right)}\cdot \left(1+Y(a)\sqrt{\frac{a}{2r_{\mathrm{e}}}}\right)\cdot {\sigma }_{\max }$$

(3)

Other variable parameters can be determined by the following equations,

$$f_{\mathrm{op}}=\{\begin{array}{ll}\max \left\{R,A_0+A_{1}R+A_2{R}^2+A_3{R}^3\right\}& 0\le R<1\\ A_0+A_{1}R& -2\le R<0\end{array}$$

(4)

$$\mathsf{\Phi }=\{\begin{array}{ll}{\left(\frac{a_{\mathrm{OL}}+r_{\mathrm{OL}}-r_{\mathrm{yi}}-r_{UL}}{a_i}\right)}^{\gamma }& a_{\mathrm{OL}}\le a_i\le a_{\mathrm{OL}}+r_{\mathrm{OL}}-r_{\mathrm{yi}}-r_{UL}\\ 1& a_i>a_{\mathrm{OL}}+r_{\mathrm{OL}}-r_{\mathrm{yi}}-r_{UL}\end{array}$$

(5)

$$K_{\mathrm{cf}}=\left[\frac{{\left(1-2v\right)}^2-\sqrt{1-v^2}}{{\left(1-2v\right)}^2-1}\cdot \frac{\pi \lambda }{{\left(1-2v\right)}^2}+\frac{\sqrt{1-v^2}-1}{{\left(1-2v\right)}^2-1}\right]\cdot K_{\mathrm{IC}}$$

(6)

$$\Delta K_{\mathrm{th}}(R)/\Delta K_{\mathrm{th}0}=\{\begin{array}{ll}{(1-R)}^{{\beta }_1}& -5\le R<0\\ {(1-R)}^{\beta }& 0\le R<0.5\\ {(1.05-1.4R+0.6R^2)}^{\beta }& 0.5\le R<1\end{array}$$

(7)

where,

• A is a material- and environmentally-sensitive constant of dimensions (MPa)⁻²;
• m is a constant representing the slope of the corresponding fatigue crack growth rate curve;
• n is the index indicating the unstable fracture;

The effect of $n$ is significant only in the unstable propagation region; a constant value of 6 is recommended for a quick and simple engineering analysis.

• K_max is the maximum stress intensity factor;
• f_op is the coefficient for the stress intensity factor at the opening level;
• Φ is the coefficient for overload/underload condition;
• ∆K_eff is the effective range of the stress intensity factor, which can be equal to K_max(1 − Φf_op);
• ∆K_effth is the effective range of the stress intensity factor at the threshold level;
• K_Cf is the fracture toughness of the material under fatigue loading;
• r_e is an empirical material constant of the inherent flaw length of the order of 1 µm;
• σ_max is the maximum applied stress;
• σ_v is an assumed “virtual strength” defined by the condition that the maximum stress of the completely uncracked (i.e., a = r_e) plain specimen;
• Y(a) is a geometrical factor;
• a is the modified crack length which is equal to r_e plus the actual crack length;
• R is the stress ratio (=σ_min/σ_max);
• a_i is the crack length in the current ith cycle;
• a_OL is the crack length when the overload cycle occurs;
• r_yi is the plastic zone in the current ith cycle;
• r_OL is the plastic zone size when the overload cycle occurs;
• r_UL is the plastic zone size when the underload cycle occurs.
• γ is a shaping exponent determined by fitting the recovering period of the experimental data;
• K_Ic is the plane strain fracture toughness of the material;
• ν is Poisson’s ratio of the material;
• λ is the coefficient to consider stress/strain state;
• ∆K_th (R) is the threshold value of stress intensity factor range under stress ratio of R;
• ∆K_th0 is the threshold value of stress intensity factor range under stress ratio of zero;

Equation (7) can be used to determine the value of ΔK_effth and the recommended values of β and β₁ are 0.3 and 0.5 respectively for steels.

Related parameters can be calculated by the follows equations,

$$\{\begin{array}{l}\frac{{\sigma }_{\mathrm{v}}}{{\sigma }_{\mathrm{b}}}=\frac{\pi }{2}\frac{1}{{\cos }^{-1}\left(\frac{1}{{\alpha }^2-1}\right)}\\ \alpha =\frac{K_{\mathrm{IC}}}{{\sigma }_{\mathrm{b}}\sqrt{\pi r_{\mathrm{e}}\left(1+\frac{Y\left(r_{\mathrm{e}}\right)}{\sqrt{2}}\right)}}>\sqrt{2}\\ \begin{array}{l}{A}_0=\left(0.825-0.34{\alpha }^{\prime }+0.05{{\alpha }^{\prime }}^2\right)\cdot {\left[\cos \left(\pi {\sigma }_{\max }/2{\sigma }_{\mathrm{fl}}\right)\right]}^{1/{\alpha }^{\prime }}\\ A_1=\left(0.415-0.071{\alpha }^{\prime }\right)\cdot {\sigma }_{\max }/{\sigma }_{\mathrm{fl}}\\ A_2=1-A_0-A_1-A_3\\ A_3=2A_0+A_1-1\\ {\sigma }_{\mathrm{fl}}=\left({\sigma }_{\mathrm{y}}+{\sigma }_{\mathrm{b}}\right)/2\end{array}\\ {\alpha }^{\prime }=\frac{1}{1-2\nu }+\frac{1-\frac{1}{1-2\nu }}{{\left[1+0.8861\cdot {\left(t^{’}/{\left(K_{\max }/{\sigma }_{\mathrm{y}}\right)}^2\right)}^{3.2251}\right]}^{0.75952}}\\ \begin{array}{l}{r}_{\mathrm{y}}=\lambda {\left(\frac{\Delta K}{{\sigma }_{\mathrm{y}}}\right)}^2\\ r_{\mathrm{OL}}=\lambda {\left(\frac{K_{\mathrm{OL}\max }}{{\sigma }_{\mathrm{y}}}\right)}^2\\ r_{\mathrm{UN}}=\lambda {\left(\frac{K_{\mathrm{UN}\max }}{{\sigma }_{\mathrm{y}}}\right)}^2\end{array}\\ \lambda =\frac{{\left(1-1.65\nu \right)}^2}{5}-\frac{1}{20n^{\prime }}{\left[{\left(1-1.65\nu \right)}^2\right]}^{\frac{1}{n^{\prime }}}+\frac{\frac{1}{\pi }-\frac{1}{2.2n^{\prime }}{\left(\frac{1}{\pi }\right)}^{\frac{1}{n^{\prime }}}-\left\{\frac{{\left(1-1.65\nu \right)}^2}{5}-\frac{1}{20n^{\prime }}{\left[{\left(1-1.65\nu \right)}^2\right]}^{\frac{1}{n^{\prime }}}\right\}}{{\left[1+\frac{t/{\left(K_{\max }/{\sigma }_{\mathrm{y}}\right)}^2}{1+1/n^{\prime }}\right]}^{1.6+1/n^{\prime }}}\end{array}$$

(8)

where,

• α^’ is the crack tip stress/strain constraint ratio, which is 1 for the plane stress state and 1/(1 − 2v) for the plane strain state.
• α is the parameter used to calculate “virtual strength”;
• σ_y is the yield strength of the material;
• σ_b is the ultimate tensile strength of the material;
• σ_fl is the flow stress of the material which can be calculated by (σ_y + σ_b)/2;
• t^’ is thickness of the cracked body;
• ∆K is stress intensity factor range;
• n^’ is material hardening index.

The main components of this model can be divided into three parts, namely, crack growth threshold condition, driving force, and unstable growth condition, as shown in Figure 3. Since the crack closure factor f_op is included in the driving force, the threshold value ∆K_th obtained in the test is also converted into the effective threshold value ∆K_effth in the driving force calculation.

2.3. The Extension to the Creep-fatigue Situation

Trapezoidal loads often exist in the loading process of underwater structures. Especially in the process of underwater equipment working after diving to a certain depth, the sustaining load lasts for relatively a long time, and its maximum stress is generally high. Under these circumstances, if the structure contains cracks, the influence of sustaining load due to creep effect on crack growth must be considered. This problem is especially important for the deep manned cabin made of titanium alloys [38,39].

The unified crack growth rate model is then extended to consider the creep effect [22]. There are two ideas when considering creep effect in trapezoidal load. One is based on cycle-by-cycle calculation, that is, not to consider the change of cracks in every load cycle. The other is to calculate crack growth in time domain, which can consider the change of cracks in a small period of time.

The extension study based on cycle-by-cycle calculation considers two aspects: linear superposition of growth rates and linear relationship between creep crack growth rate and the sustaining load time in creep stage [22]. Therefore, the additional part used to express the creep crack growth rate can be included as well as the effect of maximum stress in the expression as follows:

$${\left(\frac{\mathrm{d}a}{\mathrm{d}N}\right)}_{\mathrm{creep}}=\frac{A_2{t}_{\mathrm{hold}}{\left[\left({\sigma }_{\max }-{\sigma }_{\mathrm{R}}\right)/\left({\sigma }_{\mathrm{y}}-{\sigma }_{\mathrm{R}}\right)\right]}^k\Delta K^{m2}}{1-{\left(\raisebox{1ex}{$K_{\max }$}\!\left/ \!\raisebox{-1ex}{$K_{\mathrm{cf}}$}\right.\right)}^{n_2}}$$

(9)

When this part is added to Equation (2), the change of crack growth rate caused by creep stage in trapezoidal loading history can be considered.

where,

• A₂ is a material- and environmentally-sensitive constant of dimensions for load-maintaining phase (MPa)⁻²;
• m₂ is a constant representing the slope of the corresponding fatigue crack growth rate curve or load-maintaining phase;
• t_hold is the load-maintaining time;
• k is the material parameter for the influence level of the maximum stress;
• σ_R is the plain fatigue limit under stress ratio R.

The above cycle-by-cycle calculation model takes the amount of crack growth after each cycle as the minimum growth unit, and adds the influence of creep into it, the accuracy of which has been validated [22], but it is too rough for the situation in which a single cycle with a relatively long load-sustaining time exists. In order to ensure the safety of underwater equipment during one single operation, it is necessary to consider the crack growth within a load cycle. Under this circumstance, the da/dt model for crack growth calculation in a small time domain is suitable.

As shown in Figure 4, a single cycle is taken into account and it is assumed that only the loading and load-sustaining stages have an effect on crack growth, ignoring the effect of unloading. The model is based on the geometric relationship between the crack tip opening displacement and opening angle [40,41]. When the load increases to a reference value of σ_ref, the crack begins to propagate. At this time, the relationship between the increase of load dσ and the crack growth da can be established. When the load increases to σ_max, the crack opening displacement under creep condition [42,43] should be considered, which is different from the expression under loading condition. Based on the creep strain rate model, the relationship between crack growth da and creep time increment dt can be established [44]. The expression is shown in Equation (10). A complete crack growth curve can be obtained by superposition of crack growth under each single cycle.

$$\mathrm{d}a=\{\begin{array}{ll}\frac{1}{2}ctg(\theta )\frac{\pi }{2E{\sigma }_{\mathrm{y}}}(2\sigma ad\sigma +{\sigma }^2\mathrm{da})& {\sigma }_{\mathrm{ref}}\le \sigma <{\sigma }_{\max }\\ \frac{C{\left(\frac{\pi {\sigma }_{\max }{}^2}{E}\right)}^{\frac{n_{\mathrm{c}}}{n_{\mathrm{c}}+1}}\cdot A_{\mathrm{c}}^{{}^{\frac{1}{n_{\mathrm{c}}+1}}}{a}^{\frac{n_{\mathrm{c}}}{n_{\mathrm{c}}+1}}{t}^{\frac{1}{n_{\mathrm{c}}+1}}\left(n_{\mathrm{c}}+1\right)adt}{2\tan ({\theta }_{\max }){\left(n_{\mathrm{c}}+1\right)}^{2}at-C{\left(\frac{\pi {\sigma }_{\max }{}^2}{E}\right)}^{{}^{\frac{n_{\mathrm{c}}}{n_{\mathrm{c}}+1}}}\cdot A_{\mathrm{c}}^{{}^{\frac{1}{n_{\mathrm{c}}+1}}}{a}^{\frac{n_{\mathrm{c}}}{n_{\mathrm{c}}+1}}{t}^{\frac{1}{n_{\mathrm{c}}+1}}\cdot n_{\mathrm{c}}\left(\left(n_{\mathrm{c}}+1\right)t+\mathrm{d}t\right)}& \sigma ={\sigma }_{\max }\end{array}$$

(10)

where,

• θ is the crack tip opening angle (CTOA) and θ_max is its maximum value;
• σ_ref is a value beyond which crack starts to grow in the loading process; The value of σ_ref is calculated by assuming that the current forward plastic zone size equals to the reverse plastic zone size produced from previous loading [40,41];
• A_c and n_c are the secondary creep constant and exponent respectively;
• t is the maintaining time.

2.4. Determination of Model Parameters by Basic Data

Accurate model parameters are the basis to ensure the prediction accuracy. As shown in Figure 3, many parameters are from the basic material properties, and other parameters need to be fitted according to a set of basic crack growth rate data. In the past research, the authors’ group has accumulated some material data of A and m, some of which are published in the literature and others of which are completed in the application process. Based on these data, the model parameters that can be used in UFLP are obtained.

Table 1. Data of model parameters for common materials used in marine structures [14,45,46,47].

Material	A	m
2324-T39 Aluminum alloy [14]	2.4357 × 10−6	2.2143
6013-T651 Aluminum alloy [14]	7.6399 × 10−6	1.5997
7055-T7511 Aluminum alloy [14]	5.2495 × 10−6	1.8772
7075-T6 Aluminum alloy [14]	6.2510 × 10−6	1.9249
7075-T651 Aluminum alloy [14]	4.5774 × 10−5	1.0520
Ti-10V-Fe-3 Titanium alloy [14]	3.5880 × 10−6	1.3069
HTS-A steel [14]	1.3900 × 10−10	2.3668
300 M steel [14]	1.5770 × 10−10	2.3031
350 WT steel [14]	7.3784 × 10−10	2.0628
CrMoV [14]	3.2097 × 10−10	1.9858
Ti-6Al-4V ELI [45]	5.0000 × 10−8	3.0000
18N(250) [46]	1.5160 × 10−9	1.5400
18N(350) [46]	1.5700 × 10−9	1.4800
20MnMoNb [47]	1.1170 × 10−10	2.3979

shows the data of A and m for different metal materials.

2.5. Standardized Load-time History

The loads on marine structures, especially on ships, are very disorderly. However, the prediction of fatigue life is based on the future’s load history. Compared with the deterministic load history in the past, the randomness of assuming the future’s load is very large, which is generally based on an observation and reasonable simplification on the past loading history [17,18]. The most realistic method is to generate a standardized loading history (SLH) for practical application based on a short-term measurement as shown in Figure 5 [18].

Operating profile which describes the service conditions for the total or a representative fraction of the operating period are the basis of generating a SLH. It can be determined based on the reliable data as sea-state observations. Statistically adequate samples of load measurements under operational conditions have to be available for each load case as a basis input for relatively accurate SLH generation [17,18]. Data processing for omission of small load cycles and selection of return period length, as well as random reconstruction on them are decisive for precision. As the authors emphasized, the establishment of a standard load sequence requires the participation of multiple parties to contribute their own measurement data. Usually small load cycles are omitted according to a certain allowable filter level and then the remaining load samples in each individual operational condition can be extrapolated to appropriate time lengths based on the given structure’s operating profile combined with some extrapolation methods as the so-called Peak Over Threshold technique [18,19]. It is supposed that two kinds of time-dependent sea states including storm sea states and calm sea states make up the whole load history. A complete load history with the length of return period can be completed with storms in various random sequences. The logic of determining a sequence may depend on realistic service conditions and SLH can be determined based on the severity of fatigue crack propagation. The SLH can be used as a loading input for UFLP.

This method is more suitable for ships and offshore structures, because the loads they are subjected to are very random, and generally there is no long-term measurement data to refer to. In order to predict the future situation, it is necessary to establish a method to deal with it properly. For some underwater structures, such as the pressure hulls used in the submersibles, the load treatment is relatively simple. Each dive can be regarded as a loaded cycle. The number of diving operations per year will not be too many, then it is not as high as the loading frequency of ships and offshore structures. The depth of each dive is different, so it is also a problem of random load. In general, two processing methods can be referred to. One is to establish a random load spectrum based on diving records of existing submersibles in the past years. The other is to assume that the submersible will descend to its design depth every time, so that the random loading history will be converted into constant amplitude loading history. Usually, the design safety factor of submersible structures is relatively large, and this conservative treatment method will be affordable for the strength reserve.

3. Application Examples of UFLP

3.1. Ship and Offshore Structures

Ship and offshore structure are subjected to random loads in navigation, so the accuracy of fatigue life prediction based on crack growth depends on two aspects, i.e., the accuracy of the load model and the accuracy of crack growth rate model. Therefore, the theories introduced in Section 2.2 and Section 2.5 can provide prediction basis.

3.1.1. Ship Structures

Generally, the static analysis of the whole ship structure should be carried out first to determine the principal stress distribution, and the load should be based on the position of the local structure to be analyzed. Ref. [17] introduces the process to predict fatigue life prediction method of a typical ship structure. Hot spots can accordingly be determined as the crack-prone positions. As a typical double deck structure of a pipe-laying ship shown in Figure 6, an edge-through crack is prone to be initiated in its hot-spot position around the lighting hole.

The statistics of wave conditions can be obtained by measured data and be divided into 12 levels, all of which are considered to compose an SLH. The storm conditions in the specified sailing route can be classified into six types of storms based on the significant wave height. The average total number of storms encountered during its service life (20 years) and the mean value of each storm duration can be determined, which are respectively 93 and 3.5 days for this example [17]. Two classes including calm sea and storm sea are considered for the 12 sea states. Environmental statistics can be taken into account for determination of return period to ensure all types of storms being included in the final SLH.

According to Equation (1), the parameter Φ is used to consider load sequence effect, overload effect, or underload effect. The fatigue crack growth with effect of different storm severity can be calculated considering variable positions of some severe storms as shown in Figure 7a. The different location of storm F will affect the final calculation results.

The corresponding crack length versus load cycles (a-N) curve under each load mode can be obtained and the load sequence effect on crack growth rate is gradually decreased based on changing each storm position. The SLH can be generated after fixing the positions of all the storms. Finally, fatigue life prediction based on the curve of crack length versus voyage years can be carried out by combining the SLH and UFLP, which can provide reference to safety assessment of aged ships.

3.1.2. Offshore Structures

Offshore structures are mainly composed of welded tubular joints, in which T-joint is the most typical one and consists of chord and brace. Ref. [18] introduces the process to predict fatigue life prediction method of a typical offshore structure. The most common mode of loading it subjected to is that the brace is stretched or compressed along the axis. At this time, there will be obvious stress concentration at the junction of chord and brace. Due to the complex shape and stress concentration, the welding defects often make the junction a weak position that leads to the static strength and fatigue failure of the whole structure. The saddle point and crown point are the two weakest points, as shown in Figure 8a. There is a lot of literature related to the fatigue and fracture problem of tubular joints. In this section, a T-joint is also used as an example to demonstrate the application of the UFLP method. A surface crack is assumed to initiate in the saddle point as shown in Figure 8b. The dimensions of the tubular T-joint and the initial crack sizes are listed in

Table 2. The dimensions of the tubular T-joint and the initial crack sizes.

Parameter	D	d	T	t	L	a0	c0
Value (mm)	914.56	457.28	32	16	5486.4	3.5	10

. In the same way as dealing with the random loads of ship structure, the loads of offshore structure are still to be processed by obtaining their operating profiles and load measurement values first.

Different from the ship structures, in the 25-year service life of the offshore structures, with the change of the environment all year round, they experience the similar load history every year, so the return period can be generally considered as one year. According to the procedure introduced in Section 2.5, a standardized load-time history shown in Figure 8c with the storm sequence of A-B-D-E-C-F, which can result in the most severe condition for crack growth, can be obtained. Fatigue life prediction based on SLH and UFLP can be predicted as shown in Figure 8d. When the voyage time is 20 years, the presupposed crack will propagate to 25 mm in the thickness direction. It is safe in the whole life cycle if the failure principle of penetration is assumed. In the actual service process, the long crack may affect the stability of the whole tubular joint. Comprehensive failure modes should be considered in the overall safety assessment of the structure.

3.2. Pressure Hulls

With the development of marine technology to the deep sea [38,39], underwater pressure hull has become a typical marine structure, such as the manned cabin of a manned submersible under external pressure. The authors’ group has done a lot of research on the life-related research of deep-sea pressure hulls [47,48,49,50,51,52,53,54,55,56,57]. In addition, the deep sea environment simulator is also a kind of pressure structure (called ultra-high-pressure chamber), which is affected by internal pressure. As shown in Figure 9, the pressure hull under external pressure and the simulator under internal pressure will endure variable loading history. The load history shown in Figure 8 is the actual recorded data during the pressure test in laboratory environment. The test process is similar to the real service process, most of the pressure segments are slowly pressurized to the design depth (design pressure), and then the load is maintained for a period of time to simulate the operation process at a fixed depth, and then unload. However, in the laboratory, it may not be possible to maintain the load time to reach the underwater operation time in actual service of the equipment. As a high-pressure equipment, the material for the chamber generally has good comprehensive performance, which is usually manufactured by 20MnMoNb steel. The dimension will be increased with design pressure value. For example, the chamber in Figure 8 is with the inner diameter of 450 mm and the outer diameter of 890 mm, which is designed for simulating the full ocean depth environment with pressure of 115 MPa. As a high pressure structure used in the laboratory, its safety requirements are more stringent than other pressure vessels. Internal pressure vessel is very sensitive to crack because of the tensile stress. Fatigue life analysis based on fracture mechanics will be a satisfactory tool. As described in the previous sections, the existence of the sustaining load may affect the crack growth, but the extent of the influence is related to the performance of the material itself. To use UFLP for analysis, it needs considering the effect of sustaining load on crack growth performance of the material.

For general metal materials, if they are sensitive to sustaining load, the different crack growth rate can be obtained by adding sustaining load time of about 2 min in each cycle comparing to that under triangle loading history. The trapezoidal load sequence with sustaining time is often referred to as dwell fatigue loading. Figure 10a shows the experimental results of 20MnMoNb steel under normal fatigue loading and dwell fatigue loading [47]. As can be seen from the figure, this material is not sensitive to the effect of sustaining load. The two sets of data almost overlap. Therefore, as input data, the effect of the sustaining segment can be ignored. Equation (1) can be used for analysis. It should be emphasized that for other engineering structures, the materials they use may be particularly sensitive to the sustaining load, which needs to be examined through material test. At the same time, the basic performance parameters of the material can be obtained by tensile test and the procedure shown in Figure 3 can be referred to determine crack growth rate model parameters. Figure 10b depicts the fitted curve as well as listing model parameters. Some suggestions for fatigue analysis of pressure hulls are given in this section.

3.2.1. Fatigue Analysis of UHPC

In the fatigue analysis of this chamber based on UFLP, the second way introduced in Section 2.5 to deal with the load is adopted, that is to say, the maximum pressure is reached every time when pressing, so the variable amplitude loading can be simplified as the constant amplitude load.

Pressure hull is usually curved structure, which is more complex than ship structures. The development of finite element analysis technology saves us analysis time. In this example, the combination of ABAQUS [58] and FRANC3D [59] is applied for analysis. Figure 11 shows the joint simulation procedure. The finite element model is established by ABAQUS. The meshing is conducted as well as the constraints and load conditions applied to generate model file which can be imported into the FRANC3D. FRANC3D provides a substructure method, which divides the concerned area into sub-models, and keeps the relationship between the sub-model and the global model, as shown in Figure 12, thus reducing the calculation time. When the initial crack is introduced into the sub-model, re-meshing will be done. The sub-model with initial crack will be automatically merged with the intact part and submitted to ABAQUS for calculation for the corresponding stress intensity factors along the crack front. The crack front is renewed by selecting growth criterion as UFLP, the crack morphology can be obtained which will be the basis of the fatigue life assessment.

In the fatigue life prediction, the analysis criterion is usually set to the unstable crack growth, the crack penetrating the wall thickness, or the crack depth reaching a certain thickness. In this case, it is assumed that the chamber has reached its assumed fatigue life when the crack depth reaches one tenth of the wall thickness. This is a relatively conservative consideration. Thus, for the condition of initial crack depth c₀ = 5 mm and crack length a₀ = 3 mm, the fatigue life is about 1.11 × 10⁵ cycles.

3.2.2. Fatigue Analysis of Deep-Sea Pressure Cabin

The spherical pressure hull of manned cabin is the most important structure in deep-sea submersibles, which is generally designed with personnel access opening, several observation windows and other opening areas for penetration parts. The material must be strengthened in the vicinity of the openings and it is generally by welding a forged part. As described in the previous sections, the method based on fracture mechanics always considers the inhomogeneous materials with initial defects, and the defects in welded components are more objective. The surface crack at the weld toe is the most common defect. At the same time, the stress concentration areas are formed due to the geometric discontinuity of the welded joint surface. The spherical hull bears the alternating water pressure loading, but when the structure design is reasonable, the compressive stress produced by the external pressure inside the spherical hull is very difficult to make the crack grow. However, when it overlaps with the welding residual stress, the weld toe is prone to fatigue crack propagation and results in structural failure [57].

In engineering practice, the surface crack is generally described as a semi-elliptical crack, so a semi-elliptical surface crack at the weld toe of pressure spherical hull can be used as shown in Figure 13a. Considering the pressure spherical hull of 7000 m-class manned submersible, its maximum service load is 70 MPa. A surface crack is assumed at the weld toe of the personnel access. The material is titanium alloy with the elastic modulus E = 108 GPa and Poisson’s ratio μ = 0.3. Most of the surface cracks at the toe of the openings are parallel to the direction of the weld and are affected by the welding residual stress perpendicular to the crack surface, the value of which is about 0.3σ_s. It is assumed that the welding residual stress perpendicular to the circumferential weld is linearly distributed along the plate thickness direction, and the distribution can be simplified as shown in Figure 13b. The residual stress perpendicular to the weld direction is tensile residual stress on the outer surface of the weld. The maximum value is 0.3σ_s. The residual stress decreases gradually along the thickness direction. The residual stress at 1/4 plate thickness is 0. Then the compressive residual stress appears and increases gradually. The maximum value is 0.3σ_s at 1/2 plate thickness. The residual stress is symmetrically distributed about the middle surface of the spherical hull. The distribution form of residual stress σ^R along the plate thickness can be expressed as the following linear distribution form,

$$\{\begin{array}{ll}{\sigma }^{\mathrm{R}}(x)={\sigma }^{\mathrm{R}}(1-4x/t)& (0\le x\le t/2)\\ {\sigma }^{\mathrm{R}}(x)={\sigma }^{\mathrm{R}}(4x/t-3)& (t/2\le x\le t)\end{array}$$

(11)

Under the action of external load and residual stress, the crack will propagate. In common commercial software, Paris Law is generally used as the default crack growth rate model. A subroutine can be developed to embed UFLP into the calculation. Alternatively, the crack growth path and a series of values of stress intensity factors at the crack front can be obtained step by step, which can be as the input for life prediction by UFLP method. For underwater pressure hulls, the crack growth in thickness direction is also concerned.

The material used for the 7000 m-class manned cabin is high-strength titanium alloy, which has been proved to be affected by the load-sustaining time. Equations (9) or (10) should be used for calculation. If it is other materials, such as maraging steel [53], supplementary tests are needed to explore the effect of sustaining load. During the design, the service life of the submersible is generally considered to be 30 years, with an average of 20 dives per year. If the crack propagation is within the allowable range in the scope of 6000 cyclic loads or the maintenance cycle of 3-year maintenance, the hull is considered to be safe.

4. Summary

In this paper, the unified fatigue life prediction (UFLP) method put forward by the authors’ group is introduced, and further improvements and applications are reported. In the long-term life prediction of marine structures based on fracture mechanics, an accurate crack growth rate model is very important. The UFLP method integrates a variety of influencing factors and has a strong ability to explain various fatigue phenomena. In order to improve the prediction accuracy by considering the load sequence effect, a cycle-by-cycle integration approach is basically adopted. In order to apply the UFLP method for the design of marine structures, a concept of standard load history (SLH) for ship and offshore structures is proposed. With the development of marine technology to the deep sea, underwater pressure hull has become a typical marine structure. The deep-sea environmental simulators used to test performance of underwater equipment have also become important pressure structures. Such structures are usually subjected to cyclic sustaining loads. Therefore, the UFLP method has been extended to a model that can consider the effect of sustaining load. At the same time, in order to consider the crack growth phenomenon in a single cycle caused by long-time sustaining load, the model based on cycle-by-cycle calculation is extended to the model based on calculation in small-time domain. The two models can be transformed into each other.

However, due to the more embedded parameters, more tests are needed to determine the UFLP model parameters than the commonly used Paris law. In order to improve the convenience of application, the authors’ group also proposed some engineering determination methods of parameters through systematic research, which improves the availability of the model. At present, the fatigue calculation methods of ships and offshore structures in the classification society rules are mainly based on the cumulative damage method based on stress versys cycles (S–N) curve. However, with the improvement of lightweight requirements of the structures, the use of high-strength steel, and the fact that there are inherent defects in welded structures, it is believed that this model can provide a theoretical basis for the formulation of new evaluation rules in the future. In the continuous development, the UFLP method has become more applicable in the fatigue life prediction of marine structures.