International Scholarly Research Notices

International Scholarly Research Notices / 2011 / Article

Research Article | Open Access

Volume 2011 |Article ID 876396 | 7 pages | https://doi.org/10.5402/2011/876396

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

Academic Editor: J.-L. Marcelin
Received30 Jan 2011
Accepted27 Mar 2011
Published18 Jul 2011

Abstract

Adopt two types of damage variables, ğ‘Ž 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 ğ‘Ž 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 ğ‘Ž 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 ğ‘Ž1 and ğ‘Ž2 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 ğ‘‚î…žI′′, 𝑂I′, 𝑂1I, 𝑂2II, 𝑂3III, and 𝑂4IV and two bidirectional ordinate axis 𝑂1𝑂4 and ğ‘‚î…ž1ğ‘‚î…ž4. Between the axes ğ‘‚î…žI′′ and 𝑂1I, it is the calculation domain of the conventional material mechanics; between the axes ğ‘‚î…žI′′ and 𝑂I′, it is the calculation domain of the current ultra-high cycle fatigue; among the axes 𝑂I′, 𝑂1I, and 𝑂2II, it is the calculation domain of the damage mechanics and the microfracture mechanics; between the axes 𝑂3III and 𝑂4IV, it is calculation domain of the macrofracture mechanics; Between the axes 𝑂2II and 𝑂3III, 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 ğ‘‘ğ‘Ž/𝑑𝑁 (that can also carve up the damage evolving rate 𝑑𝐷1/𝑑𝑁1 or short crack growth rate ğ‘‘ğ‘Ž1/𝑑𝑁1 at crack forming stage or the damage evolving rate 𝑑𝐷2/𝑑𝑁2 or long crack growth rate ğ‘‘ğ‘Ž2/𝑑𝑁2 at crack growth stage), and the downward direction is presented as each stage life 2N. The distance ğ‘‚î…žğ‘‚ between axis ğ‘‚î…žI′′ and 𝑂I′ is shown as the region of the nominal stress 𝑆 or remote stress ğœŽğ‘œ; the distance ğ‘‚î…žğ‘‚2 between axis ğ‘‚î…žI′′ and 𝑂2II is shown as the region from uncrack to microcrack initiation; the distance 𝑂2𝑂3 between axes 𝑂2II and 𝑂3III is shown as the region relative to life 𝑁oimic-mac from microcrack growth to macrocrack forming. Consequently, the distance ğ‘‚î…žğ‘‚3 is shown as the region relating to life 𝑁mac from grains size to microcrack initiation until macrocrack forming; the distance ğ‘‚î…žğ‘‚4 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 ğ‘‚î…žğ‘‚4 and abscissa axes 𝑂I′, 𝑂1I, and 𝑂2II is presented to be the relationship between the damage evolving rate 𝑑𝐷1/𝑑𝑁1 (or the short crack growth rate ğ‘‘ğ‘Ž1/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 ğ‘‚î…žğ‘‚4 and 𝑂3III (𝑂4IV) 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 Δ𝛿𝑡/2 (ğ‘‘ğ‘Ž2/𝑑𝑁2-Δ𝐾/2, Δ𝐽/2 and Δ𝛿𝑡/2) at macro-crack growth stage. The coordinate system combined from downward ordinate axis 𝑂4𝑂1 and abscissa axes 𝑂2II, and 𝑂3III is presented as the relationship between the Δ𝐻/2-, Δ𝐾/2-amplitude, and the life 2N (or between the Δ𝜀𝑝/2-, Δ𝛿𝑡/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 𝑂3III, point 𝐴1 is corresponding to fatigue strength coefficient ğœŽî…žğ‘“; point 𝐶1 is corresponding to fatigue ductility coefficient ğœ€î…žğ‘“; point 𝐹 is corresponding to ultra-high cycle fatigue strength coefficient ğœŽî…žuhf. 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 ğ‘‘ğ‘Ž1/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 ğ‘‘ğ‘Ž1/𝑑𝑁1 and Δ𝐼/2; inverted 𝐶1𝐵𝐶 shows the relation between the Δ𝜀𝑝/2-2N. And the curve 𝐴1𝐴2 at crack growth stage is showed as under high cycle loading: positive direction 𝐴1𝐴2; shows ğ‘‘ğ‘Ž2/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 ğ‘‘ğ‘Ž2/𝑑𝑁2 and Δ𝛿𝑡/2 under low-cycle loading, inverted 𝐶2𝐶1, shows the relation between Δ𝛿𝑡/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).

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 ğ‘‚î…žğ¼î…žî…ž and ğ‘‚ğ¼î…ž 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 mechanicsΔ𝐺uh=Î”ğœŽuhâ‹…ğ‘Žğœ‡1/𝑛,Î”ğºî…žuh=Î”ğœŽuh⋅𝐷𝜇1/𝑛,(1) where the Δ𝐺uh and Î”ğºî…žuh 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 ğ‘Žğœ‡(𝐷𝜇) which is originated from interior materials. And the mathematic models of driving force between the axes 𝑂1I and 𝑂3III are the stress intensity factor range Δ𝐻1 of shortcrack or the damage stress intensity factor range Î”ğ»î…ž1.Δ𝐻1=Î”ğœŽ1⋅𝑚1âˆšğ‘Ž1,(2) or Î”ğ»î…ž1=Î”ğœŽ1⋅𝑚1√𝐷1(3) that is calculation domain of the microfracture mechanics or damage mechanics. And the mathematic models of driving force between the axes 𝑂3III and 𝑂4IV are the stress intensity factor range Δ𝐾2 of longcrack or the damage stress intensity factor range Î”ğ¾î…ž2 at the second stage Δ𝐾2√=Î”ğœŽâ‹…ğœ‹ğ‘Ž2,(4) or Î”ğ¾î…ž2√=Î”ğœŽâ‹…ğœ‹ğ·2,(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.


Forward directionReverse direction

Ubiety area of triangleGeometrical meaning of edge vectorPhysical meaning of edge vectorGeometrical meaning of edge vectorPhysical meaning of edge vector
Triangle between the axes 𝑂  𝐼   and 𝑂 𝐼  Level edgeDriving forceLevel edgeDriving force
Vertical edgeDamage rateVertical edgeLife of one cycle
Bevel edge (slope)ElastoplasticBevel vector (slope)Elastoplastic
Area of trianglePower (w)AreaWork (w) in one cycle

Triangle between the axes 𝑂 1 I and 𝑂 3 I I I Level edgeDriving forceLevel edgeDriving force
Vertical edgeDamage rateVertical edgeLife of one cycle
Bevel edge (slope)ElastoplasticBevel vector (slope)Elastoplastic
Area of trianglePower (w)Area of triangleWork (w) in one cycle

Triangle between the axes 𝑂 3 I I I and 𝑂 4 I V Level edgeDriving forceLevel edgeDriving force
Vertical edgeDamage rateVertical edgeLife of one cycle
Bevel edge (slope)Elastoplastic and stiffnessBevel vector (slope)Elastoplastic and stiffness
Area of trianglePowerArea of triangleWork in one cycle

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:𝐾Imax≅0.65ğœŽ0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=𝑦1ğœŽâ‹…ğ‘š1âˆšğ‘Ž1≤𝐻mac𝑚1√𝑚⋅MPa,(7) where 𝐻mac=ğœŽâ‹…ğ‘š1âˆšğ‘Žmac𝑚1√𝑚⋅MPa(8)𝐻mac= Critical stress intensity factor of short crack that is corresponding to macro-crack size at about the threshold level Δ𝐾th 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]ğ»î…ž=𝑦1ğœŽâ‹…ğ‘š1√𝐷1â‰¤ğ»î…žmac,MPa𝑚1√𝐷1orMPa,(9) where ğ»î…žmac=ğœŽâ‹…ğ‘š1√𝐷macMPa𝑚1√𝐷1orMPa(10)ğ»î…žmac: Critical damage stress intensity factor that is equivalent to the 𝐻mac. 𝐷mac: Critical damage level that is equivalent to ğ‘Žmac; if ğ‘Žmac=0.7mm, then 𝐷mac=0.7.

Conversion method between variables ğ‘Ž 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 ğ‘Žmac=0.7mm, then critical damage value of the equivalent is 𝐷mac=0.7 damage unit. And define that ğ‘Ž0<ğ‘Ž1â‰¤ğ‘Žmac; 𝐷0<𝐷1≤𝐷mac. The 𝐷0 is an initial damage value corresponding to micro-crack size ğ‘Ž0, 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]: 𝐾I=𝐾2=𝑦2âˆšâ‹…ğœŽğœ‹ğ‘Ž2<𝐾I𝑐=𝐾2𝑐√𝑚⋅MPa,(11)𝐾I𝑐=𝐾2𝑐√=ğœŽğœ‹â‹…ğ‘Ž2𝑐,√𝑚⋅MPa,(12) where 𝐾I𝑐(=𝐾2𝑐)= 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]ğ¾î…ž2=𝑦2âˆšâ‹…ğœŽğœ‹ğ·2<ğ¾î…ž2𝑐,MPa𝑚1√𝐷1𝐾orMPa,(13)2𝑐√=ğœŽğœ‹ğ·2𝑐,√MPa𝐷2orMPa(14)ğ¾î…ž2𝑐 is a critical damage stress intensity factor equivalent to the 𝐾I𝑐(𝐾2𝑐). 𝐷2𝑐 is a critical damage value correspondent to the critical crack size ğ‘Ž2𝑐.

Conversion method between variables ğ‘Ž2 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 ğ»î…ž and the ğ¾î…ž2 at each stage become identical with the unit of stress because the dimensional unit between the variables ğ‘Ž 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., ğ‘Ž1=ğ‘Ž2=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 ğœŽğ‘=545MPa, yield limit ğœŽğ‘¦=349MPa, the strain-hardening exponent 𝑛=0.136, fatigue strength exponent in short crack growth 𝑚1=11.478, threshold level Δ𝐾th√=6.87MPa𝑚, critical stress intensity factor 𝐾1𝑐√=97.3MPa𝑚=𝐾2𝑐=𝐾Ic√=97.3(MPa𝑚), critical damage stress intensity factor ğ¾î…ž2𝑐√=97.3MPa𝐷2 equivalent to the 𝐾I𝑐(𝐾2𝑐), the critical stress intensity factor 𝐻mac=350𝑚1√𝑚 and the damage stress intensity factor ğ»î…žmac=350(MPa𝑚1√𝐷1) of short crack corresponding to the threshold level Δ𝐾th, and the mean sizes ğ‘Ž0=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 ğ»î…ž at short crack size ğ‘Ž0=100𝜇m undergone after damage for a crystal grain and the stress intensity factor 𝐾2=𝐾I and the damage stress intensity factor ğ¾î…ž2 at long crack size ğ‘Ž2=2mm. Computing approach is as follows.

5.1. Computing the Stress Intensity Factor 𝐻 and the Damage Stress Intensity Factor ğ»î…ž for Short Crack
5.1.1. Computing the Stress Intensity Factor 𝐻 for Short Crack

According to (7), select the computing parameter: ğ‘Ž1=100𝜇m=1.0−4m; the local stress is 840 MPa at focal point of stress undergone damage; If we take 𝑦1=1.1, 𝐻1=𝑦1ğœŽâ‹…ğ‘š1âˆšğ‘Ž1=1.1×840×11.478√100𝜇m=1.1×840×11.478√1.0−4m=414.41𝑚1√𝑚.(15) This results in 𝐻1=414.41𝑚1√𝑚>ğ»î…žmac=350𝑚1√𝑚MPa.(16) So this short crack is growth.

5.1.2. Computing of the Damage Stress Intensity Factor ğ»î…ž for Short Crack

According to (9), when the ğ‘Ž1=100𝜇m=1.0−4m, equivalent damage value of short crack is the 𝐷1=100=1.0−4 (damage unit).

So 𝐻1=𝑦1Ã—ğœŽâ‹…ğ‘š1√𝐷1=1.1×840×11.478√100=1.1×840×11.478√1.0−4=414.41MPa𝑚1√𝐷or(MPa).(17) This results in 𝐻1=414.41MPa𝑚1√𝐷>ğ»î…žmac=350(MPa).(18) So this short crack is also growth.

5.2. Computing the Stress Intensity Factor 𝐾2=𝐾𝐼 and the Damage Stress Intensity Factor ğ¾î…ž2 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, ğ‘Ž2=2mm, working stress ğœŽ=280MPa;𝐾2=𝐾I=𝑦2âˆšÃ—ğœŽÃ—ğœ‹â‹…ğ‘Ž2√=1.05×280×√𝜋×0.002=23.30MPa𝑚.(19) This results in 𝐾2=23.30<ğ¾î…ž2𝑐=𝐾I𝑐√=97.3MPa𝑚.(20) So the pressure vessel is safe.

5.2.2. Computing the Damage Stress Intensity Factor ğ¾î…ž2 for Long Crack

According to equation (13), take the computing parameter 𝑦2=1.05; when take ğ‘Ž2=2mm, the equivalent damage value of long crack is 𝐷2=2000=2.0−3 (damage unit).

So 𝐾2=𝐾I=𝑦2âˆšÃ—ğœŽğœ‹â‹…ğ·2√=1.05×280×√𝜋×0.002=23.30MPa𝐷or(MPa).(21) This results in 𝐾2=23.30<ğ¾î…ž2𝑐=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 ğ‘Ž1 and ğ‘Ž2 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 ğ‘ğ‘ğ‘Žğµğ´1𝐴2 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(ğ»î…ž1) to the 𝐾2(ğ¾î…ž2). 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 𝐾I but the driving force of microdamage at crack forming stage is defined by the author as the damage stress intensity factor ğ»î…ž1, 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 ğ»î…ž1 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⋅𝑚1√𝑚, and the unit of the ğ»î…ž1 is the MPa𝑚1√𝐷1. Both the units are different, but the unit of the damage stress intensity factor ğ»î…ž1 is the same as the unit of the stress, and the 𝐻1 and the ğ»î…ž1 are all essentially the stress intensity factor and a relation of equivalents. And the stress intensity factor 𝐾2 and the damage stress intensity factor ğ¾î…ž2 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 ğ¾î…ž2 is the √MPa⋅𝐷2. Both the units are also different, but the unit of the damage stress intensity factor ğ¾î…ž2 is also the same as the unit of the stress, and the 𝐾2 and the ğ¾î…ž2 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, 𝐷𝜇, 𝐷1 = microdamage variable equivalent to micro-crack ğ‘Žğœ‡ or short crack ğ‘Ž1 at the crack forming stage (first stage), and 𝐷2 = damage variable at the crack growth stage (the second stage).(2)ğ‘Žğœ‡, ğ‘Ž1 = micro-, short-crack size at the crack forming stage (variable of the first stage), ğ‘Ž2 = macro-, long-crack size at the crack growth stage (variable of the second stage), ğ‘Ž10 = initial size of micro-crack forming (ordaining value), crack size ğ‘Žth = corresponding to the threshold level Δ𝐾th, original size ğ‘Ž20 = ğ‘Žmac of macro-crack forming stage (ordaining value), and ğ‘Ž2𝑐 = critical size of long crack. (3)Î”ğœŽ = nominal stress range, Î”ğœŽ0 = remote stress range, Î”ğœŽ/2 = stress amplitude, Δ𝜀𝑝 = strain range, Δ𝜀𝑝/2 = strain amplitude, and ğœŽğ‘š = mean stress. (4)Δ𝐺 or Î”ğºî…ž = micro-crack stress intensity factor range or microdamage stress intensity factor range corresponding to microcrack size ğ‘Žğœ‡ or microdamage 𝐷𝜇 under ultra-high cycle fatigue, 𝐻1 = stress intensity factor of short crack, 𝐻mac = critical stress intensity factor of short crack, ğ»î…ž1 = damage stress intensity factor of short crack and Î”ğ»î…ž1 or Δ𝐻1 or Δ𝐻1/2 = stress intensity factor range or stress intensity factor amplitude relative to short-crack ğ‘Ž1; Î”ğ»î…ž1/2 = 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 ğ‘Ž1. (5)ğ‘Žî…ž1 = fatigue strength exponent under ultra-high cycle fatigue, ğ‘î…ž1 = fatigue strength exponent under high cycle fatigue, ğ‘î…ž1 = fatigue ductility exponent under low cycle fatigue, 𝑛1= fatigue strength exponent in micro-crack growth rate equation under ultra-high cycle fatigue, 𝑛1=−1/ğ‘Žî…ž1; 𝑚1 = fatigue strength exponent in short-crack growth rate equation under high-cycle fatigue, 𝑚1=−1/ğ‘î…ž1, ğ‘šî…ž1 = fatigue ductility exponent in short-crack growth rate equation under low cycle fatigue, ğ‘šî…ž1=−1/ğ‘î…ž1, ğ‘î…ž2 = fatigue strength exponent of the macro-crack growth stage under high-cycle fatigue, ğ‘î…ž2 = 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/ğ‘î…ž2, ğ‘šî…ž2 = the fatigue ductility exponent in crack growth rate equation under low-cycle fatigue, ğ‘šî…ž2=−1/ğ‘î…ž2.(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, ğ‘‘ğ‘Ž/𝑑𝑁 = crack growth rate, ğ‘‘ğ‘Ž1/𝑑𝑁1 = short crack growth rate at the crack forming stage, and ğ‘‘ğ‘Ž2/𝑑𝑁2 = its rate at the macro-crack growth stage.(7)𝑁oi = life of correspondance to medial damage variable 𝐷oi or short-crack medial size ğ‘Žoi at the first stage, and 𝑁oj = life of correspondance to medial damage variable 𝐷oj or long-crack medial size ğ‘Žoj at the second stage.(8)𝐾2=𝐾I= stress intensity factor of long crack, ğ¾î…ž2 = damage stress intensity factor of long crack, 𝐽-integral of long crack; and crack tip opening displacement of long crack, 𝐾𝑚= mean stress intensity factor, Δ𝐾/2 = stress intensity factor amplitude of correspondance to macro-crackâ€‰â€‰ğ‘Ž2,Δ𝐽/2 = 𝐽-integral amplitude corresponding to macro-crack ğ‘Ž2, Δ𝛿𝑡/2 = crack tip opening displacement amplitude corresponding to macro-crack ğ‘Ž2.(9)𝐾1𝑐 = critical stress intensity factor corresponding to macro-crack critical ğ‘Ž2𝑐, 𝐾eff = effective stress intensity factor to be applicable in Paris’s equation, 𝐽𝑐 = critical 𝐽-integral value corresponding to macro-crack critical ğ‘Ž2𝑐, and 𝛿𝑐 = critical crack tip opening displacement corresponding to macro-crack critical ğ‘Ž2𝑐.

Acknowledgment

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

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Copyright © 2011 Yangui Yu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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