The Computing of Intersectant Relations for Its Strength Problem on Damage and Fracture to Materials with Short and Long Crack

Abstract

Adopt two types of damage variables, 𝑎 and 𝐷, and make the bidirectional combined coordinate system and the bidirectional curves in the whole process, describing their damage evolutive behaviors on fatigue damage-fracture to elastic-plastic steels; communicate the cross-referencing between their computing models and between describing curves at each stage among varied disciplines; bring forward a viewpoint about the driving force of material damage, that is the damage stress factor at crack forming stage, Provide the computation expressions and computing methods of the strength problem of materials with short crack and long crack at each stage; reveal the geometrical and the physical meaning of force triangle and its edge vector at each stage, Provide the conversion methods between the variables, the equations, the material constants and the dimensional units; Indicate the physical and the geometrical meanings for some key parameters. This will be having practical significance for promoting developing, and applying each discipline.

1. Introduction

As is now well known, we adopt the crack size 𝑎 as a variable in the fracture mechanics to describe crack growth process undergoing damage for a material and we adopt a damage variable 𝐷 in the damage mechanics to describe an evolutive process undergoing damage for one. Either the sign 𝑎 or the sign 𝐷 are all virtually damage variables so, we could also adopt the damage 𝐷 to describe the evolutive law of a structure material with crack. References [1–3] had made out such research. In each discipline we have all in-house features and its advantages. If we can communicate and convert corresponding relations for that among the damage variables, and the equations, the material constants, the dimensional units which describe the material behavior for varied discipline and provide some conversion methods, thus we are also able to adopt the same variables 𝐷₁ and 𝐷₂ or the variable 𝑎₁ and 𝑎₂ to compute the strength and the life at each stage or even in overall process for structures and materials undergoing fatigue damage [4, 5]. And the conventional materials and damage mechanics are made by inheritance and development and the modern one is made by all better combination and application. Based on this aim THE authors adopt the mathematical derivation and computational analyses used with computer by long-range research, educe a series of the computation expressions and the computing methods. Thus, this may be having practical significance for promoting, developing, and applying of some disciplines.

2. Bidirectional Combined Coordinate System and Bidirectional Curves in the Whole Process

In some of branch disciplines on fatigue-damage fracture, for finding their correlations among variables, curves, equations, and material constants of describing material behaviors at each stage and for connecting their relations to each other, we must put up analysis and developments for a number of problems that are above mentioned. Here, it is by means of bidirectional combined coordinate system that is, adopted in Figure 1 [6] we express the damage evolving process of material behavior at each stage and in the whole course, which consists of six abscissa axes 𝑂^_I′′, 𝑂_I^′, 𝑂_1I, 𝑂_2II, 𝑂_3III, and 𝑂_4IV and two bidirectional ordinate axis 𝑂₁𝑂₄ and 𝑂^1𝑂^4. Between the axes 𝑂^_I′′ and 𝑂_1I, it is the calculation domain of the conventional material mechanics; between the axes 𝑂^_I′′ and 𝑂_I^′, it is the calculation domain of the current ultra-high cycle fatigue; among the axes 𝑂_I^′, 𝑂_1I, and 𝑂_2II, it is the calculation domain of the damage mechanics and the microfracture mechanics; between the axes 𝑂_3III and 𝑂_4IV, it is calculation domain of the macrofracture mechanics; Between the axes 𝑂_2II and 𝑂_3III, it is all applicable calculation domain for the microfracture mechanics and macrofracture mechanics. The upward direction along the ordinate axis is presented as damage evolving rate 𝑑𝐷/𝑑𝑁 or crack growth rate 𝑑𝑎/𝑑𝑁 (that can also carve up the damage evolving rate 𝑑𝐷₁/𝑑𝑁₁ or short crack growth rate 𝑑𝑎₁/𝑑𝑁₁ at crack forming stage or the damage evolving rate 𝑑𝐷₂/𝑑𝑁₂ or long crack growth rate 𝑑𝑎₂/𝑑𝑁₂ at crack growth stage), and the downward direction is presented as each stage life 2N. The distance 𝑂^𝑂 between axis 𝑂^_I′′ and 𝑂_I^′ is shown as the region of the nominal stress 𝑆 or remote stress 𝜎_𝑜; the distance 𝑂^𝑂₂ between axis 𝑂^_I′′ and 𝑂_2II is shown as the region from uncrack to microcrack initiation; the distance 𝑂₂𝑂₃ between axes 𝑂_2II and 𝑂_3III is shown as the region relative to life 𝑁_oimic-mac from microcrack growth to macrocrack forming. Consequently, the distance 𝑂^𝑂₃ is shown as the region relating to life 𝑁_mac from grains size to microcrack initiation until macrocrack forming; the distance 𝑂^𝑂₄ is shown as the region relating to the lifelong life 2N from micro-crack initiation until fracture of structure material. The coordinate system combined from upward axis 𝑂^𝑂₄ and abscissa axes 𝑂_I^′, 𝑂_1I, and 𝑂_2II is presented to be the relationship between the damage evolving rate 𝑑𝐷₁/𝑑𝑁₁ (or the short crack growth rate 𝑑𝑎₁/𝑑𝑁₁) and the damage stress factor amplitude Δ𝐻/2 (or damage strain factor amplitude Δ𝐼/2) at crack forming stage; the coordinate system combined from 𝑂^𝑂₄ and 𝑂_3III (𝑂_4IV) at the same direction is presented to be the relationship between macrocrack growth rate and stress intensity factor amplitude Δ𝐾/2, 𝐽-integral amplitude Δ𝐽/2, and crack tip displacement amplitude Δ𝛿_𝑡/2 (𝑑𝑎₂/𝑑𝑁₂-Δ𝐾/2, Δ𝐽/2 and Δ𝛿_𝑡/2) at macro-crack growth stage. The coordinate system combined from downward ordinate axis 𝑂₄𝑂₁ and abscissa axes 𝑂_2II, and 𝑂_3III is presented as the relationship between the Δ𝐻/2-, Δ𝐾/2-amplitude, and the life 2N (or between the Δ𝜀_𝑝/2-, Δ𝛿_𝑡/2-amplitude, and the life 2N). The curve abcd is the ultra-high cycle fatigue one to correspond to stress below fatigue limit. On abscissa 𝑂_3III, point 𝐴₁ is corresponding to fatigue strength coefficient 𝜎^𝑓; point 𝐶₁ is corresponding to fatigue ductility coefficient 𝜀^𝑓; point 𝐹 is corresponding to ultra-high cycle fatigue strength coefficient 𝜎^_uhf. The 𝐴𝐵𝐴₁ shows the varying regularities of elastic material behaviors as under high cycle loading at macro-crack forming stage; positive direction 𝐴𝐵𝐴₁ shows the relation between 𝑑𝐷₁/𝑑𝑁₁ (or 𝑑𝑎₁/𝑑𝑁₁) and Δ𝐻/2; inverted 𝐴₁𝐵𝐴 shows the relation between the Δ𝐻/2 and 2N. The curve 𝐶𝐵𝐶₁ shows the varying regularities of plastic material behaviors, as is under low-cycle loading at macro-crack forming stage; positive direction 𝐶𝐵𝐶₁ shows the relation between 𝑑𝑎₁/𝑑𝑁₁ and Δ𝐼/2; inverted 𝐶₁𝐵𝐶 shows the relation between the Δ𝜀_𝑝/2-2N. And the curve 𝐴₁𝐴₂ at crack growth stage is showed as under high cycle loading: positive direction 𝐴₁𝐴₂; shows 𝑑𝑎₂/𝑑𝑁₂-Δ𝐾/2 (Δ𝐽/2); inverted 𝐴₂𝐴₁, shows the relation between the Δ𝐾/2 and Δ𝐽/2-2N. The 𝐶₁𝐶₂ shows the positive direction relation between the 𝑑𝑎₂/𝑑𝑁₂ and Δ𝛿_𝑡/2 under low-cycle loading, inverted 𝐶₂𝐶₁, shows the relation between Δ𝛿_𝑡/2 (Δ𝐽/2) and 2N. And it should point that the 𝐴𝐴₁𝐴₂ (curve 11′) is expressed for the curve under symmetrical cycle loading (i.e., under zero mean stress); the 𝐷𝐷₁𝐷₂ (curve 33′) is expressed for the curve under unsymmetrical cycle loading (i.e., under nonzero mean stress).

Figure 1 

Bidirectional combined coordinate system and bidirectional curves in the whole process.

3. Relations among Force Triangles and Relations among Edge Vectors of Force Triangle Itself at Each Stage

Due to extent of material undergone damage at each stage is different and due to the variance in the material behaviors after undergoing damage, the rigidity of material is also different to ensue from change, so the force triangles consist of edge vectors at different stages which are also varied relationships among force triangles, and relationships among the edge vectors themselves at each stage, their geometrical and physical meanings are all compiled in Table 1. We can make out from the force triangle in each area from Figure 1 find the mathematic model of driving force is different in each discipline. The driving force seen from force triangle between the axes 𝑂^𝐼^ and 𝑂𝐼^ is the damage stress intensity factor under ultra-high cycle fatigue. We should also point out that this is the calculation domain in micro-damage mechanics or in conventional material mechanicsΔΔ𝐺𝐺_uhuh=Δ𝜎_uh⋅𝑎^𝜇1/𝑛,=Δ𝜎_uh⋅𝐷^𝜇1/𝑛,(1) where the Δ𝐺_uh and Δ𝐺^_uh are, respectively, the microcrock stress intensity factor or the microdamage stress intensity factor under ultra-high cycle fatigue to correspond a crystal grain or microdamage defect 𝑎_𝜇(𝐷_𝜇) which is originated from interior materials. And the mathematic models of driving force between the axes 𝑂_1I and 𝑂_3III are the stress intensity factor range Δ𝐻₁ of shortcrack or the damage stress intensity factor range Δ𝐻^1.Δ𝐻₁=Δ𝜎₁⋅√^𝑚1𝑎₁,(2) or Δ𝐻^1=Δ𝜎₁⋅√^𝑚1𝐷₁(3) that is calculation domain of the microfracture mechanics or damage mechanics. And the mathematic models of driving force between the axes 𝑂_3III and 𝑂_4IV are the stress intensity factor range Δ𝐾₂ of longcrack or the damage stress intensity factor range Δ𝐾^2 at the second stage Δ𝐾₂=Δ𝜎⋅√𝜋𝑎₂,(4) or Δ𝐾^2=Δ𝜎⋅√𝜋𝐷₂,(5) that is calculation domain of the macro-fracture mechanics. Relations among force triangles can be expressly made out in Figure 1; the relations among edge vectors of force triangle iteself and their geometrical and physical meaning at each stage are all compiled in Table 1.

Table 1 

Relations among the edge vectors of the force triangle iteself and their geometrical and physical meanings at each stage.

4. Computing the Damage Strength for Elastic-Plastic Material under High-Cycle Fatigue

Based upon above-mentioned viewpoints we bring forward the computation expressions and calculation methods of strength for elastic-plastic steels which are the material behaviors undergoing fatigue damage at different stages. And we proportionally analyze for the cross-referencing between their equations, material constants, and dimensional units and also explain conversion methods between them.

4.1. Computing of Strength for Material with Crack

4.1.1. Computing of Strength at the First Stage

About computing the material strength with short crack, many researchers had presented varied computing models and made out valuable contributions [7, 8]. Murakami [9] provides the computation expression as follows:𝐾_Imax≅0.65𝜎₀𝜋√area.(6) And research and analysis of the driving force of force triangle in Figure 1 and according to above explanation and the present author provide again a computational model to describe stress-strain about the tip of short crack that is the stress intensity factor at the first stage as following [10–12]:𝐻₁=𝑦₁𝜎⋅√^𝑚1𝑎₁≤𝐻_mac√^𝑚1𝑚⋅MPa,(7) where 𝐻_mac=𝜎⋅√^𝑚1𝑎_mac√^𝑚1𝑚⋅MPa(8)𝐻_mac= Critical stress intensity factor of short crack that is corresponding to macro-crack size at about the threshold level Δ𝐾_th and more than one, that is artificially definite according to the size of structure member.

𝑦₁ is a correction coefficient concerned with shape and size, for example, of a structure member. And it became the damage stress intensity factor at the same stage when we adopt damage variable 𝐷₁ to describe [13]𝐻^=𝑦₁𝜎⋅√^𝑚1𝐷₁≤𝐻^_mac,MPa√^𝑚1𝐷₁orMPa,(9) where 𝐻^_mac=𝜎⋅√^𝑚1𝐷_macMPa√^𝑚1𝐷₁orMPa(10)𝐻^_mac: Critical damage stress intensity factor that is equivalent to the 𝐻_mac. 𝐷_mac: Critical damage level that is equivalent to 𝑎_mac; if 𝑎_mac=0.7mm, then 𝐷_mac=0.7.

Conversion method between variables 𝑎 and 𝐷 and between the dimensional units defines 1 mm equivalent to one damage unit (nondimensional value), 1 m equivalent to 1000 damage unit. If critical 𝑎_mac=0.7mm, then critical damage value of the equivalent is 𝐷_mac=0.7 damage unit. And define that 𝑎₀<𝑎₁≤𝑎_mac; 𝐷₀<𝐷₁≤𝐷_mac. The 𝐷₀ is an initial damage value corresponding to micro-crack size 𝑎₀, and it is advised to take the average value of 10 crystal sizes or to take the maximum size of crystal size. It should also be pointed out that the stresses are all local ones in (2)–(5) and (7)–(10).

4.1.2. Computing of Strength at the Second Stage

The computational model to describe stress strain about the tip of long crack at the second stage had been provided by famous scientists Broek and Hellan as follows [14, 15]: 𝐾_I=𝐾₂=𝑦₂⋅𝜎√𝜋𝑎₂<𝐾_I𝑐=𝐾_2𝑐√𝑚⋅MPa,(11)𝐾_I𝑐=𝐾_2𝑐=𝜎√𝜋⋅𝑎_2𝑐,√𝑚⋅MPa,(12) where 𝐾_I𝑐(=𝐾_2𝑐)= Critical stress intensity factor of long crack 𝑦₂ is a correctional coefficient concerned with the shape of crack and shape and size of structure member. And it becomes the damage stress intensity factor at the same stage if we adopt damage variable 𝐷₂ to describe [16, 17]𝐾^2=𝐾𝑦₂⋅𝜎√𝜋𝐷₂<𝐾^2𝑐,MPa√^𝑚1𝐷₁orMPa,(13)_2𝑐=𝜎√𝜋𝐷_2𝑐,MPa√𝐷₂orMPa(14)𝐾^2𝑐 is a critical damage stress intensity factor equivalent to the 𝐾_I𝑐(𝐾_2𝑐). 𝐷_2𝑐 is a critical damage value correspondent to the critical crack size 𝑎_2𝑐.

Conversion method between variables 𝑎₂ and 𝐷₂ and between the dimensional units at the second stage is the same as the first stage. It should be pointed out that the dimensional unit of the damage stress intensity factor 𝐻^ and the 𝐾^2 at each stage become identical with the unit of stress because the dimensional unit between the variables 𝑎 and 𝐷 is converted.

It should be pointed out that both the values between the stress intensity factor of short-crack and the stress intensity of long crack are different under the condition of the same crack size (e.g., 𝑎₁=𝑎₂=0.1mm) because the mechanisms of material damage and because their mathematical models and dimensional units of both driving forces are all different. But both growth rates at the turning point from short crack to long crack should be accordant or near.

5. Computing Example

A pressure vessel adopts the steel 16 MnR to make its strength limit of material 𝜎_𝑏=545MPa, yield limit 𝜎_𝑦=349MPa, the strain-hardening exponent 𝑛=0.136, fatigue strength exponent in short crack growth 𝑚₁=11.478, threshold level Δ𝐾_th=6.87MPa√𝑚, critical stress intensity factor 𝐾_1𝑐=97.3MPa√𝑚=𝐾_2𝑐=𝐾_Ic=97.3(MPa√𝑚), critical damage stress intensity factor 𝐾^2𝑐=97.3MPa√𝐷₂ equivalent to the 𝐾_I𝑐(𝐾_2𝑐), the critical stress intensity factor 𝐻_mac=350√^𝑚1𝑚 and the damage stress intensity factor 𝐻^_mac=350(MPa√^𝑚1𝐷₁) of short crack corresponding to the threshold level Δ𝐾_th, and the mean sizes 𝑎₀=20∼30𝜇m of crystal grains; its working stress is 280MPa of pressure vessel and local stress is 840MPa at focal point of stress. Try respectively, to compute the stress intensity factor 𝐻 and the damage stress intensity factor 𝐻^ at short crack size 𝑎₀=100𝜇m undergone after damage for a crystal grain and the stress intensity factor 𝐾₂=𝐾_I and the damage stress intensity factor 𝐾^2 at long crack size 𝑎₂=2mm. Computing approach is as follows.

5.1. Computing the Stress Intensity Factor 𝐻 and the Damage Stress Intensity Factor 𝐻^ for Short Crack

5.1.1. Computing the Stress Intensity Factor 𝐻 for Short Crack

According to (7), select the computing parameter: 𝑎₁=100𝜇m=1.0⁻⁴m; the local stress is 840 MPa at focal point of stress undergone damage; If we take 𝑦₁=1.1, 𝐻₁==1𝑦₁.1𝜎×⋅8√^𝑚140𝑎×₁=1._11.4781×840×√^11.478100𝜇m=414.41_𝑚1√1.0⁻⁴m√𝑚.(15) This results in 𝐻₁=414.41√^𝑚1𝑚>𝐻^_mac=350√^𝑚1𝑚MPa.(16) So this short crack is growth.

5.1.2. Computing of the Damage Stress Intensity Factor 𝐻^ for Short Crack

According to (9), when the 𝑎₁=100𝜇m=1.0⁻⁴m, equivalent damage value of short crack is the 𝐷₁=100=1.0⁻⁴ (damage unit).

So 𝐻₁=𝑦₁×𝜎⋅√^𝑚1𝐷₁==11..11××884400××√^11.478_11.478100√1.0⁻⁴=414.41MPa√^𝑚1𝐷or(MPa).(17) This results in 𝐻₁=414.41MPa√^𝑚1𝐷>𝐻^_mac=350(MPa).(18) So this short crack is also growth.

5.2. Computing the Stress Intensity Factor 𝐾₂=𝐾_𝐼 and the Damage Stress Intensity Factor 𝐾^2 for Long Crack

5.2.1. Computing the Stress Intensity Factor 𝐾₂ for Long Crack

According to (11), select a parameter, take the correction coefficient 𝑦₂=1.05, 𝑎₂=2mm, working stress 𝜎=280MPa;𝐾₂=𝐾_I=𝑦₂×𝜎×√𝜋⋅𝑎₂==12.30.530×2M80P×a√√𝜋×0.002𝑚.(19) This results in 𝐾₂=23.30<𝐾^2𝑐=𝐾_I𝑐=97.3MPa√𝑚.(20) So the pressure vessel is safe.

5.2.2. Computing the Damage Stress Intensity Factor 𝐾^2 for Long Crack

According to equation (13), take the computing parameter 𝑦₂=1.05; when take 𝑎₂=2mm, the equivalent damage value of long crack is 𝐷₂=2000=2.0⁻³ (damage unit).

So 𝐾₂=𝐾_I=𝑦₂×𝜎√𝜋⋅𝐷₂==21.30.350×2M80P×a√√𝜋×0.002𝐷or(MPa).(21) This results in 𝐾₂=23.30<𝐾^2𝑐=97.3(MPa).(22) So the pressure vessel is also safe.

6. Summarization

The bidirectional combined coordinate system, the bidirectional curves in the whole process and their force triangles at each stage are important scientific method and tool to communicate the cross-referencing which is to describe the evolutive process of a material behavior undergoing of the fatigue damage in each of the disciplines, which are able to make available communications and conversions for those complicated correlations among some variables, some equations, curves, and dimensional units, which are clearely able to explain the geometrical and physical meanings for the key parameters. Thus, it is also able to adopt same variables 𝐷₁ and 𝐷₂ or the variables 𝑎₁ and 𝑎₂ to compute the strength and the lift at each stage or even in overall process for structures and materials undergoing a fatigue damage. And the conventional material and damage mechanics are able of making inheritance and development, and the modern one are all able to make better combination and application. Thus, that may be having practical significance for promoting developing, and applying some disciplines.

7. Conclusions

(1)About the Problem of the Relation between the Mathematical Model and the Material Behavior. Under identical loading, when the structure material is undergoing fatigue damage, the differences between the mathematical models to describe the material behavior are due to the degree of damage undergone at varied stages that makes the stiffness of the material change, which find that expressions in the curves 𝑐𝑏𝑎𝐵𝐴₁𝐴₂ in evolutive process are turned to take place at the points b, B, and 𝐴₁. The slopes of the curves at each stage also are brought to change. The exponents in the equation also became from the 𝑚₁=−1/𝑏₁ at crack forming stage to the 𝑚₂=−1/𝑏₂ at crack growth stage. The driving forces became from the 𝐻₁(𝐻^1) to the 𝐾₂(𝐾^2). The compositive material constants became from the 𝐴₁ to the 𝐴₂.(2)About the Problem of Driving Force. It well known that the driving force of long crack growth in macro fracture mechanics is the stress intensity factor 𝐾_I but the driving force of microdamage at crack forming stage is defined by the author as the damage stress intensity factor 𝐻^1, and the driving force in microfracture mechanics is defined as the stress intensity factor 𝐻₁ of short-crack growth.(3)About Problem of Dimensional Units. The stress intensity factor 𝐻₁ and the damage stress intensity factor 𝐻^1 of short crack at crack forming stage are all to describe the stress strain about the crack tip of micro-crack, the unit of the 𝐻₁ is MPa⋅√^𝑚1𝑚, and the unit of the 𝐻^1 is the MPa√^𝑚1𝐷₁. Both the units are different, but the unit of the damage stress intensity factor 𝐻^1 is the same as the unit of the stress, and the 𝐻₁ and the 𝐻^1 are all essentially the stress intensity factor and a relation of equivalents. And the stress intensity factor 𝐾₂ and the damage stress intensity factor 𝐾^2 of long crack at crack growth stage are all to describe the stress strain about the crack tip of long crack, the unit of the 𝐾₂ is MPa⋅√𝑚, and the unit of 𝐾^2 is the MPa⋅√𝐷₂. Both the units are also different, but the unit of the damage stress intensity factor 𝐾^2 is also the same as the unit of the stress, and the 𝐾₂ and the 𝐾^2 are all essentially the stress intensity factor and a relation between equivalents. In addition, both values and dimensional units between the stress intensity factor of short crack and the stress intensity factor of long crack are also different under the condition of same crack size.

Nomenclature

(1)𝐷 = damage variable in the whole process, 𝐷_𝜇, 𝐷₁ = microdamage variable equivalent to micro-crack 𝑎_𝜇 or short crack 𝑎₁ at the crack forming stage (first stage), and 𝐷₂ = damage variable at the crack growth stage (the second stage).(2)𝑎_𝜇, 𝑎₁ = micro-, short-crack size at the crack forming stage (variable of the first stage), 𝑎₂ = macro-, long-crack size at the crack growth stage (variable of the second stage), 𝑎₁₀ = initial size of micro-crack forming (ordaining value), crack size 𝑎_th = corresponding to the threshold level Δ𝐾_th, original size 𝑎₂₀ = 𝑎_mac of macro-crack forming stage (ordaining value), and 𝑎_2𝑐 = critical size of long crack. (3)Δ𝜎 = nominal stress range, Δ𝜎₀ = remote stress range, Δ𝜎/2 = stress amplitude, Δ𝜀_𝑝 = strain range, Δ𝜀_𝑝/2 = strain amplitude, and 𝜎_𝑚 = mean stress. (4)Δ𝐺 or Δ𝐺^ = micro-crack stress intensity factor range or microdamage stress intensity factor range corresponding to microcrack size 𝑎_𝜇 or microdamage 𝐷_𝜇 under ultra-high cycle fatigue, 𝐻₁ = stress intensity factor of short crack, 𝐻_mac = critical stress intensity factor of short crack, 𝐻^1 = damage stress intensity factor of short crack and Δ𝐻^1 or Δ𝐻₁ or Δ𝐻₁/2 = stress intensity factor range or stress intensity factor amplitude relative to short-crack 𝑎₁; Δ𝐻^1/2 = damage stress intensity factor range or damage stress intensity factor amplitude relative to damage variable 𝐷₁.

Δ𝐼 or Δ𝐼/2 = damage strain factor range or damage strain factor amplitude relative to short crack 𝑎₁. (5)𝑎^1 = fatigue strength exponent under ultra-high cycle fatigue, 𝑏^1 = fatigue strength exponent under high cycle fatigue, 𝑐^1 = fatigue ductility exponent under low cycle fatigue, 𝑛₁= fatigue strength exponent in micro-crack growth rate equation under ultra-high cycle fatigue, 𝑛₁=−1/𝑎^1; 𝑚₁ = fatigue strength exponent in short-crack growth rate equation under high-cycle fatigue, 𝑚₁=−1/𝑏^1, 𝑚^1 = fatigue ductility exponent in short-crack growth rate equation under low cycle fatigue, 𝑚^1=−1/𝑐^1, 𝑏^2 = fatigue strength exponent of the macro-crack growth stage under high-cycle fatigue, 𝑐^2 = the fatigue ductility exponent at the macro-crack growth stage under low-cycle fatigue, 𝑚₂ = the fatigue strength exponent in crack growth rate equation under high-cycle fatigue, 𝑚₂=−1/𝑏^2, 𝑚^2 = the fatigue ductility exponent in crack growth rate equation under low-cycle fatigue, 𝑚^2=−1/𝑐^2.(6)𝑑𝐷/𝑑𝑁 = damage evolutive rate, 𝑑𝐷₁/𝑑𝑁₁ = damage evolutive rate at the crack forming stage, 𝑑𝐷₂/𝑑𝑁₂ = damage evolutive rate at the macro-crack growth stage, 𝑑𝑎/𝑑𝑁 = crack growth rate, 𝑑𝑎₁/𝑑𝑁₁ = short crack growth rate at the crack forming stage, and 𝑑𝑎₂/𝑑𝑁₂ = its rate at the macro-crack growth stage.(7)𝑁_oi = life of correspondance to medial damage variable 𝐷_oi or short-crack medial size 𝑎_oi at the first stage, and 𝑁_oj = life of correspondance to medial damage variable 𝐷_oj or long-crack medial size 𝑎_oj at the second stage.(8)𝐾₂=𝐾_I= stress intensity factor of long crack, 𝐾^2 = damage stress intensity factor of long crack, 𝐽-integral of long crack; and crack tip opening displacement of long crack, 𝐾_𝑚= mean stress intensity factor, Δ𝐾/2 = stress intensity factor amplitude of correspondance to macro-crack  𝑎₂,Δ𝐽/2 = 𝐽-integral amplitude corresponding to macro-crack 𝑎₂, Δ𝛿_𝑡/2 = crack tip opening displacement amplitude corresponding to macro-crack 𝑎₂.(9)𝐾_1𝑐 = critical stress intensity factor corresponding to macro-crack critical 𝑎_2𝑐, 𝐾_eff = effective stress intensity factor to be applicable in Paris’s equation, 𝐽_𝑐 = critical 𝐽-integral value corresponding to macro-crack critical 𝑎_2𝑐, and 𝛿_𝑐 = critical crack tip opening displacement corresponding to macro-crack critical 𝑎_2𝑐.

Acknowledgment

The Authors thank sincerey the Zhejiang Province Natural Science fund Committee that gave support and provided subsidization of the research funds.