#### Abstract

Adopt two types of damage variables, and , 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 as a variable in the fracture mechanics to describe crack growth process undergoing damage for a material and we adopt a damage variable in the damage mechanics to describe an evolutive process undergoing damage for one. Either the sign or the sign are all virtually damage variables so, we could also adopt the damage to describe the evolutive law of a structure material with crack. References [1β3] 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 and or the variable and 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 , , , , , and and two bidirectional ordinate axis and . Between the axes and , it is the calculation domain of the conventional material mechanics; between the axes and , it is the calculation domain of the current ultra-high cycle fatigue; among the axes , , and , it is the calculation domain of the damage mechanics and the microfracture mechanics; between the axes and , it is calculation domain of the macrofracture mechanics; Between the axes and , 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 or crack growth rate (that can also carve up the damage evolving rate or short crack growth rate at crack forming stage or the damage evolving rate or long crack growth rate at crack growth stage), and the downward direction is presented as each stage life 2N. The distance between axis and is shown as the region of the nominal stress or remote stress ; the distance between axis and is shown as the region from uncrack to microcrack initiation; the distance between axes and is shown as the region relative to life from microcrack growth to macrocrack forming. Consequently, the distance is shown as the region relating to life from grains size to microcrack initiation until macrocrack forming; the distance 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 and abscissa axes , , and is presented to be the relationship between the damage evolving rate (or the short crack growth rate ) and the damage stress factor amplitude (or damage strain factor amplitude ) at crack forming stage; the coordinate system combined from and () at the same direction is presented to be the relationship between macrocrack growth rate and stress intensity factor amplitude , -integral amplitude , and crack tip displacement amplitude (-, and ) at macro-crack growth stage. The coordinate system combined from downward ordinate axis and abscissa axes , and is presented as the relationship between the -, -amplitude, and the life 2N (or between the -, -amplitude, and the life 2N). The curve abcd is the ultra-high cycle fatigue one to correspond to stress below fatigue limit. On abscissa , point is corresponding to fatigue strength coefficient ; point is corresponding to fatigue ductility coefficient ; point is corresponding to ultra-high cycle fatigue strength coefficient . The shows the varying regularities of elastic material behaviors as under high cycle loading at macro-crack forming stage; positive direction shows the relation between (or ) and ; inverted shows the relation between the and . The curve shows the varying regularities of plastic material behaviors, as is under low-cycle loading at macro-crack forming stage; positive direction shows the relation between and ; inverted shows the relation between the . And the curve at crack growth stage is showed as under high cycle loading: positive direction ; shows - (); inverted , shows the relation between the and . The shows the positive direction relation between the and under low-cycle loading, inverted , shows the relation between () and 2N. And it should point that the (curve 11β²) is expressed for the curve under symmetrical cycle loading (i.e., under zero mean stress); the (curve 33β²) 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 and 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 where the and 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 which is originated from interior materials. And the mathematic models of driving force between the axes and are the stress intensity factor range of shortcrack or the damage stress intensity factor range . or that is calculation domain of the microfracture mechanics or damage mechanics. And the mathematic models of driving force between the axes and are the stress intensity factor range of longcrack or the damage stress intensity factor range at the second stage or 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.

#### 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: 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 [10β12]: where Critical stress intensity factor of short crack that is corresponding to macro-crack size at about the threshold level and more than one, that is artificially definite according to the size of structure member.

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 to describe [13] where : Critical damage stress intensity factor that is equivalent to the . : Critical damage level that is equivalent to ; if , then .

Conversion method between variables and and between the dimensional units defines 1βmm equivalent to one damage unit (nondimensional value), 1βm equivalent to 1000 damage unit. If critical , then critical damage value of the equivalent is damage unit. And define that ; . The is an initial damage value corresponding to micro-crack size , 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)β(5) and (7)β(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]: where Critical stress intensity factor of long crack 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 to describe [16, 17] is a critical damage stress intensity factor equivalent to the . is a critical damage value correspondent to the critical crack size .

Conversion method between variables and 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 and the at each stage become identical with the unit of stress because the dimensional unit between the variables and 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., ) 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 16βMnR to make its strength limit of material , yield limit , the strain-hardening exponent , fatigue strength exponent in short crack growth , threshold level , critical stress intensity factor , critical damage stress intensity factor equivalent to the , the critical stress intensity factor and the damage stress intensity factor of short crack corresponding to the threshold level , and the mean sizes of crystal grains; its working stress is of pressure vessel and local stress is at focal point of stress. Try respectively, to compute the stress intensity factor and the damage stress intensity factor at short crack size undergone after damage for a crystal grain and the stress intensity factor and the damage stress intensity factor at long crack size . Computing approach is as follows.

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

###### 5.1.1. Computing the Stress Intensity Factor for Short Crack

According to (7), select the computing parameter: ; the local stress is 840βMPa at focal point of stress undergone damage; If we take , This results in So this short crack is growth.

###### 5.1.2. Computing of the Damage Stress Intensity Factor for Short Crack

According to (9), when the , equivalent damage value of short crack is the (damage unit).

So This results in So this short crack is also growth.

##### 5.2. Computing the Stress Intensity Factor and the Damage Stress Intensity Factor for Long Crack

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

According to (11), select a parameter, take the correction coefficient , , working stress ; This results in So the pressure vessel is safe.

###### 5.2.2. Computing the Damage Stress Intensity Factor for Long Crack

According to equation (13), take the computing parameter ; when take , the equivalent damage value of long crack is (damage unit).

So This results in 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 and or the variables and 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

*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 in evolutive process are turned to take place at the points

*b*,

*B*, and . The slopes of the curves at each stage also are brought to change. The exponents in the equation also became from the at crack forming stage to the at crack growth stage. The driving forces became from the () to the (). The compositive material constants became from the to the .

*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 but the driving force of microdamage at crack forming stage is defined by the author as the damage stress intensity factor , and the driving force in microfracture mechanics is defined as the stress intensity factor of short-crack growth.

*About Problem of Dimensional Units.*The stress intensity factor and the damage stress intensity factor 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 is , and the unit of the is the . Both the units are different, but the unit of the damage stress intensity factor is the same as the unit of the stress, and the and the are all essentially the stress intensity factor and a relation of equivalents. And the stress intensity factor and the damage stress intensity factor 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 is , and the unit of is the . Both the units are also different, but the unit of the damage stress intensity factor is also the same as the unit of the stress, and the and the 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

= damage variable in the whole process, , = microdamage variable equivalent to micro-crack or short crack at the crack forming stage (first stage), and = damage variable at the crack growth stage (the second stage)., = micro-, short-crack size at the crack forming stage (variable of the first stage), = macro-, long-crack size at the crack growth stage (variable of the second stage), = initial size of micro-crack forming (ordaining value), crack size = corresponding to the threshold level , original size = of macro-crack forming stage (ordaining value), and = critical size of long crack. = nominal stress range, = remote stress range, = stress amplitude, = strain range, = strain amplitude, and = mean stress. or = micro-crack stress intensity factor range or microdamage stress intensity factor range corresponding to microcrack size or microdamage under ultra-high cycle fatigue, = stress intensity factor of short crack, = critical stress intensity factor of short crack, = damage stress intensity factor of short crack and or or = stress intensity factor range or stress intensity factor amplitude relative to short-crack ; = damage stress intensity factor range or damage stress intensity factor amplitude relative to damage variable .or = damage strain factor range or damage strain factor amplitude relative to short crack . = fatigue strength exponent under ultra-high cycle fatigue, = fatigue strength exponent under high cycle fatigue, = fatigue ductility exponent under low cycle fatigue, = fatigue strength exponent in micro-crack growth rate equation under ultra-high cycle fatigue, ; = fatigue strength exponent in short-crack growth rate equation under high-cycle fatigue, , = fatigue ductility exponent in short-crack growth rate equation under low cycle fatigue, , = fatigue strength exponent of the macro-crack growth stage under high-cycle fatigue, = the fatigue ductility exponent at the macro-crack growth stage under low-cycle fatigue, = the fatigue strength exponent in crack growth rate equation under high-cycle fatigue, , = the fatigue ductility exponent in crack growth rate equation under low-cycle fatigue, . = damage evolutive rate, = damage evolutive rate at the crack forming stage, = damage evolutive rate at the macro-crack growth stage, = crack growth rate, = short crack growth rate at the crack forming stage, and = its rate at the macro-crack growth stage. = life of correspondance to medial damage variable or short-crack medial size at the first stage, and = life of correspondance to medial damage variable or long-crack medial size at the second stage. stress intensity factor of long crack, = damage stress intensity factor of long crack, -integral of long crack; and crack tip opening displacement of long crack, mean stress intensity factor, = stress intensity factor amplitude of correspondance to macro-crackββ, = -integral amplitude corresponding to macro-crack , = crack tip opening displacement amplitude corresponding to macro-crack . = critical stress intensity factor corresponding to macro-crack critical , = effective stress intensity factor to be applicable in Parisβs equation, = critical -integral value corresponding to macro-crack critical , and = critical crack tip opening displacement corresponding to macro-crack critical .

#### Acknowledgment

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