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

Authors: Yangui Yu; Yiming Sun; Yanghui Ma; Feng Xu
Journal: ISRN Mechanical Engineering
 (2011)

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## 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.


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## Body

## 1. Introduction

As is now well known, we adopt the crack sizea 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 Figure1 [6] we express the damage evolving process of material behavior at each stage and in the whole course, which consists of six abscissa axes OI′′′, OI′, O1I, O2II, O3III, and O4IV and two bidirectional ordinate axis O1O4 and O1′O4′. Between the axes OI′′′ and O1I, it is the calculation domain of the conventional material mechanics; between the axes OI′′′ 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 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 OI′′′ and OI′ is shown as the region of the nominal stress S or remote stress σo;  the distance O′O2 between axis OI′′′ 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 O′O3 is shown as the region relating to life Nmac 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 OI′, O1I, and O2II 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 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 (da2/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 σf′; point C1 is corresponding to fatigue ductility coefficient εf′; point F is corresponding to ultra-high cycle fatigue strength coefficient σuhf′. 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 Δεp/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 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).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 Table1. 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 mechanics(1)ΔGuh=Δσuh⋅aμ1/n,ΔGuh′=Δσuh⋅Dμ1/n,
where the ΔGuh and ΔGuh′ 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 aμ(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 ΔH1′.(2)ΔH1=Δσ1⋅a1m1,
or
(3)ΔH1′=Δσ1⋅D1m1
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 ΔK2′ at the second stage (4)ΔK2=Δσ⋅πa2,
or
(5)ΔK2′=Δσ⋅πD2,
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.
Forward directionReverse directionUbiety area of triangleGeometrical meaning of edge vectorPhysical meaning of edge vectorGeometrical meaning of edge vectorPhysical meaning of edge vectorTriangle between the axesO′I′′ and OI′Level edgeDriving forceLevel edgeDriving forceVertical edgeDamage rateVertical edgeLife of one cycleBevel edge (slope)ElastoplasticBevel vector (slope)ElastoplasticArea of trianglePower (w)AreaWork (w) in one  cycleTriangle between the axesO1I and O3IIILevel edgeDriving forceLevel edgeDriving forceVertical edgeDamage rateVertical edgeLife of one cycleBevel edge (slope)ElastoplasticBevel vector (slope)ElastoplasticArea of trianglePower (w)Area of triangleWork (w) in one  cycleTriangle between the axesO3III and O4IVLevel edgeDriving forceLevel edgeDriving forceVertical edgeDamage rateVertical edgeLife of one cycleBevel edge (slope)Elastoplastic and stiffnessBevel vector (slope)Elastoplastic and stiffnessArea 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:(6)KImax≅0.65σ0πarea.
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]:(7)H1=y1σ⋅a1m1≤Hmac(mm1⋅MPa),
where   (8)Hmac=σ⋅amacm1(mm1⋅MPa)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](9)H′=y1σ⋅D1m1≤Hmac′,(MPaD1m1)orMPa,
where
(10)Hmac′=σ⋅Dmacm1(MPaD1m1)orMPaHmac′: Critical damage stress intensity factor that is equivalent to the Hmac. Dmac: Critical damage level that is equivalent to amac; if amac=0.7mm, then Dmac=0.7.Conversion method between variablesa 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 amac=0.7mm, then critical damage value of the equivalent is Dmac=0.7 damage unit. And define that a0<a1≤amac; D0<D1≤Dmac. 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]: (11)KI=(K2)=y2⋅σπa2<KIc(=K2c)(m⋅MPa),(12)KIc=K2c=σπ⋅a2c,(m⋅MPa),
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](13)K2′=y2⋅σπD2<K2c′,(MPaD1m1)orMPa,(14)K2c′=σπD2c,(MPaD2)orMPaK2c′ is a critical damage stress intensity factor equivalent to the KIc(K2c). D2c is a critical damage value correspondent to the critical crack size a2c.Conversion method between variablesa2 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 K2′ 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.


## 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:(6)KImax≅0.65σ0πarea.
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]:(7)H1=y1σ⋅a1m1≤Hmac(mm1⋅MPa),
where   (8)Hmac=σ⋅amacm1(mm1⋅MPa)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](9)H′=y1σ⋅D1m1≤Hmac′,(MPaD1m1)orMPa,
where
(10)Hmac′=σ⋅Dmacm1(MPaD1m1)orMPaHmac′: Critical damage stress intensity factor that is equivalent to the Hmac. Dmac: Critical damage level that is equivalent to amac; if amac=0.7mm, then Dmac=0.7.Conversion method between variablesa 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 amac=0.7mm, then critical damage value of the equivalent is Dmac=0.7 damage unit. And define that a0<a1≤amac; D0<D1≤Dmac. 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]: (11)KI=(K2)=y2⋅σπa2<KIc(=K2c)(m⋅MPa),(12)KIc=K2c=σπ⋅a2c,(m⋅MPa),
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](13)K2′=y2⋅σπD2<K2c′,(MPaD1m1)orMPa,(14)K2c′=σπD2c,(MPaD2)orMPaK2c′ is a critical damage stress intensity factor equivalent to the KIc(K2c). D2c is a critical damage value correspondent to the critical crack size a2c.Conversion method between variablesa2 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 K2′ 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.


## 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:(6)KImax≅0.65σ0πarea.
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]:(7)H1=y1σ⋅a1m1≤Hmac(mm1⋅MPa),
where   (8)Hmac=σ⋅amacm1(mm1⋅MPa)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](9)H′=y1σ⋅D1m1≤Hmac′,(MPaD1m1)orMPa,
where
(10)Hmac′=σ⋅Dmacm1(MPaD1m1)orMPaHmac′: Critical damage stress intensity factor that is equivalent to the Hmac. Dmac: Critical damage level that is equivalent to amac; if amac=0.7mm, then Dmac=0.7.Conversion method between variablesa 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 amac=0.7mm, then critical damage value of the equivalent is Dmac=0.7 damage unit. And define that a0<a1≤amac; D0<D1≤Dmac. 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]: (11)KI=(K2)=y2⋅σπa2<KIc(=K2c)(m⋅MPa),(12)KIc=K2c=σπ⋅a2c,(m⋅MPa),
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](13)K2′=y2⋅σπD2<K2c′,(MPaD1m1)orMPa,(14)K2c′=σπD2c,(MPaD2)orMPaK2c′ is a critical damage stress intensity factor equivalent to the KIc(K2c). D2c is a critical damage value correspondent to the critical crack size a2c.Conversion method between variablesa2 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 K2′ 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 materialσb=545MPa, yield limit σy=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 K2c′=97.3MPaD2 equivalent to the KIc(K2c), the critical stress intensity factor Hmac=350mm1 and the damage stress intensity factor Hmac′=350(MPaD1m1) of short crack corresponding to the threshold level ΔKth, 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=KI and the damage stress intensity factor K2′ at long crack size a2=2mm. Computing approach is as follows.


### 5.1. Computing the Stress Intensity FactorH and the Damage Stress Intensity Factor  H′ for Short Crack

#### 5.1.1. Computing the Stress Intensity FactorH 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,  (15)H1=y1σ⋅a1m1=1.1×840×100μm11.478=1.1×840×1.0-4m11.478=414.41(mm1).
This results in
(16)H1=414.41(mm1)>Hmac′=350(mm1MPa).
So this short crack is growth.


#### 5.1.2. Computing of the Damage Stress Intensity FactorH′ 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(17)H1=y1×σ⋅D1m1=1.1×840×10011.478=1.1×840×1.0-411.478=414.41(MPaDm1)or(MPa).
This results in
(18)H1=414.41(MPaDm1>Hmac′)=350(MPa).
So this short crack is also growth.


### 5.2. Computing the Stress Intensity FactorK2=KI and the Damage Stress Intensity Factor K2′ for Long Crack

#### 5.2.1. Computing the Stress Intensity FactorK2 for Long Crack

According to (11), select a parameter, take the correction coefficient y2=1.05, a2=2mm, working stress σ=280MPa;(19)K2=KI=y2×σ×π⋅a2=1.05×280×π×0.002=23.30(MPam).
This results in
(20)K2=23.30<K2c′=KIc=97.3(MPam).
So the pressure vessel is safe.


#### 5.2.2. Computing the Damage Stress Intensity FactorK2′ 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(21)K2=KI=y2×σπ⋅D2=1.05×280×π×0.002=23.30(MPaD)or(MPa).
This results in
(22)K2=23.30<K2c′=97.3(MPa).
So the pressure vessel is also safe.


## 5.1. Computing the Stress Intensity FactorH and the Damage Stress Intensity Factor  H′ for Short Crack

### 5.1.1. Computing the Stress Intensity FactorH 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,  (15)H1=y1σ⋅a1m1=1.1×840×100μm11.478=1.1×840×1.0-4m11.478=414.41(mm1).
This results in
(16)H1=414.41(mm1)>Hmac′=350(mm1MPa).
So this short crack is growth.


### 5.1.2. Computing of the Damage Stress Intensity FactorH′ 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(17)H1=y1×σ⋅D1m1=1.1×840×10011.478=1.1×840×1.0-411.478=414.41(MPaDm1)or(MPa).
This results in
(18)H1=414.41(MPaDm1>Hmac′)=350(MPa).
So this short crack is also growth.


## 5.1.1. Computing the Stress Intensity FactorH 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,  (15)H1=y1σ⋅a1m1=1.1×840×100μm11.478=1.1×840×1.0-4m11.478=414.41(mm1).
This results in
(16)H1=414.41(mm1)>Hmac′=350(mm1MPa).
So this short crack is growth.


## 5.1.2. Computing of the Damage Stress Intensity FactorH′ 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(17)H1=y1×σ⋅D1m1=1.1×840×10011.478=1.1×840×1.0-411.478=414.41(MPaDm1)or(MPa).
This results in
(18)H1=414.41(MPaDm1>Hmac′)=350(MPa).
So this short crack is also growth.


## 5.2. Computing the Stress Intensity FactorK2=KI and the Damage Stress Intensity Factor K2′ for Long Crack

### 5.2.1. Computing the Stress Intensity FactorK2 for Long Crack

According to (11), select a parameter, take the correction coefficient y2=1.05, a2=2mm, working stress σ=280MPa;(19)K2=KI=y2×σ×π⋅a2=1.05×280×π×0.002=23.30(MPam).
This results in
(20)K2=23.30<K2c′=KIc=97.3(MPam).
So the pressure vessel is safe.


### 5.2.2. Computing the Damage Stress Intensity FactorK2′ 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(21)K2=KI=y2×σπ⋅D2=1.05×280×π×0.002=23.30(MPaD)or(MPa).
This results in
(22)K2=23.30<K2c′=97.3(MPa).
So the pressure vessel is also safe.


## 5.2.1. Computing the Stress Intensity FactorK2 for Long Crack

According to (11), select a parameter, take the correction coefficient y2=1.05, a2=2mm, working stress σ=280MPa;(19)K2=KI=y2×σ×π⋅a2=1.05×280×π×0.002=23.30(MPam).
This results in
(20)K2=23.30<K2c′=KIc=97.3(MPam).
So the pressure vessel is safe.


## 5.2.2. Computing the Damage Stress Intensity FactorK2′ 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(21)K2=KI=y2×σπ⋅D2=1.05×280×π×0.002=23.30(MPaD)or(MPa).
This results in
(22)K2=23.30<K2c′=97.3(MPa).
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 variablesD1 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(H1′) to the K2(K2′). 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 H1′, 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 H1′ 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·mm1, and the unit of the H1′ is the MPaD1m1. Both the units are different, but the unit of the damage stress intensity factor H1′ is the same as the unit of the stress, and the H1 and the H1′ are all essentially the stress intensity factor and a relation of equivalents. And the stress intensity factor K2 and the damage stress intensity factor K2′ 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 K2′ is the MPa·D2. Both the units are also different, but the unit of the damage stress intensity factor K2′ is also the same as the unit of the stress, and the K2 and the K2′ 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 aμ or short crack a1 at the crack forming stage (first stage), and D2 = damage variable at the crack growth stage (the second stage).(2)
aμ, 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 ath = corresponding to the threshold level ΔKth, original size a20 = amac of macro-crack forming stage (ordaining value), and a2c = critical size of long crack.(3)
Δσ = nominal stress range, Δσ0 = remote stress range, Δσ/2 = stress amplitude, Δεp = strain range, Δεp/2 = strain amplitude, and σm = mean stress.(4)
ΔG or ΔG′ = micro-crack stress intensity factor range or microdamage stress intensity factor range corresponding to microcrack size aμ or microdamage Dμ under ultra-high cycle fatigue, H1 = stress intensity factor of short crack, Hmac = critical stress intensity factor of short crack, H1′ = damage stress intensity factor of short crack and ΔH1′ or ΔH1 or ΔH1/2 = stress intensity factor range or stress intensity factor amplitude relative to short-crack a1; ΔH1′/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)
a1′ = fatigue strength exponent under ultra-high cycle fatigue, b1′ = fatigue strength exponent under high cycle fatigue, c1′ = fatigue ductility exponent under low cycle fatigue,  n1= fatigue strength exponent in micro-crack growth rate equation under ultra-high cycle fatigue, n1=-1/a1′; m1 = fatigue strength exponent in short-crack growth rate equation under high-cycle fatigue, m1=-1/b1′, m1′ = fatigue ductility exponent in short-crack growth rate equation under low cycle fatigue, m1′=-1/c1′, b2′ = fatigue strength exponent of the macro-crack growth stage under high-cycle fatigue, c2′ = 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/b2′, m2′ = the fatigue ductility exponent in crack growth rate equation under low-cycle fatigue, m2′=-1/c2′.(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)
Noi = life of correspondance to medial damage variable Doi or short-crack medial size aoi at the first stage, and Noj = life of correspondance to medial damage variable Doj or long-crack medial size aoj at the second stage.(8)
K2=KI= stress intensity factor of long crack, K2′ = 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-crack  a2,ΔJ/2 = J-integral amplitude corresponding to macro-crack a2, Δδt/2 = crack tip opening displacement amplitude corresponding to macro-crack a2.(9)
K1c = critical stress intensity factor corresponding to macro-crack critical a2c, Keff = effective stress intensity factor to be applicable in Paris’s equation, Jc = critical J-integral value corresponding to macro-crack critical a2c, and δc = critical crack tip opening displacement corresponding to macro-crack critical a2c.



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*Source: 876396-2011-07-18.xml*