International Journal of Materials Science and Applications
Volume 4, Issue 3, May 2015, Pages: 173-188

The Life Predicting Calculations Based on Conventional Material Constants from Short Crack to Long Crack Growth Process

Yangui Yu

Zhejiang GuangXin New Technology Application Academy of Electromechanical and Chemical Engineering, Hangzhou, China

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To cite this article:

Yangui Yu. The Life Predicting Calculations Based on Conventional Material Constants from Short Crack to Long Crack Growth Process. International Journal of Materials Science and Applications. Vol. 4, No. 3, 2015, pp. 173-188. doi: 10.11648/j.ijmsa.20150403.15


Abstract: To use the theoretical approach, to adopt the multiplication-method of two-parameters, by means of the traditional and the modern material constants, thereby to establish some of new calculation models in all crack growth process. In which are the equations of the driving forces, the crack-growth-rate-linking-equation in whole process, and the life predictions; and to propose yet some calculating expressions under different loading conditions. For key material parameters give their new concepts, and provide new functional formulas, define their physical and geometrical meanings. For the transition crack size from micro to macro crack growth process, provide concrete calculation processes and methods. Thereby realize the lifetime predicting calculations in whole process based on conventional materials constants and by the multiplication method of two parameters.

Keywords: Short Crack and Long Crack, Calculating Modeling, Lifetime Prediction, High Cycle Fatigue, Low Cycle Fatigue


1. Introduction

As everyone knows for the traditional material mechanics, that is a calculable subject, and it has made valuable contributions for every industrial engineering designs and calculations. But it cannot accurately calculate the life problems for some structures when it is used for pre-existing flaws and under repeated loading. In that it has no to contain such calculable parameters in its calculating models as the crack variable  or as the damage variable . But, for the fracture mechanics or the damage mechanics, due to there are these variables, they can all calculate above problems. However, nowadays latter these disciplines are all mainly depended on tests by fatigue, damage and fracture.

Author studies their compositions of traditional mathematical models and modern models for above disciplines; and researches crack growth behaviors for some steels of containing micro and macro flaws, thinks that in the mechanics and the engineering fields, also to exist such scientific principles of genetic and clone technology as life science [1]. Author has done some of works used the theoretical approach as above the similar principles [1-8]. For example, for some strength calculation models from micro to macro crack had been provided by reference [2], for some rate calculation models from micro to macro damage (or crack) growth had been proposed by references [3-8] which were all models in each stage even in whole process, and to be under different loading conditions. Two years ago, in order to do the lifetime calculations in whole process on fatigue-damage-fracture for an engineering structure, author was by means of Google Scholar to search the lifetime prediction models, as had been no found for this kind of calculation equations. After then, author bases on the comprehensive figure 1 of material behaviors has provided some models as in references [2-3]; Recently sequentially complemented for it [1], and continues to research this item, still applies above genetic principles to study and analyze data in references, thereby to provide another some new calculable models for the crack growth driving force and for the lifetime predictions. The purpose is to try to make the fracture mechanics step by step become calculable discipline as the traditional material mechanics. That way, may be having practical significances for decrease experiments, for promoting engineering applying and developing for relevant disciplines.

2. The Life Prediction Calculations for Elastic-Plastic Steels Containing Pre-Flaws

Under repeating load, for some elastic-plastic steels containing pre-cracks, their crack growth rate and life predicting calculations for short crack growth and micro-damage growth process had been calculated by means of some methods in reference [9-10], that are the -factor method by two-parameters multiplication  with stress and strain; For the long crack growth process, this paper also adopts similar to above method, that is the -factor method by the two-parameters multiplication with the stress intensity factor and the crack tip open displacement ; Moreover, here to provide another a new calculating method---the stress ()-method. Thereby achieve severally the purpose for life’s prediction calculations in different stage.

Figure 1. Comprehensive figure of material behaviors (Bidirectional combined coordinate system and simplified schematic curves in the whole process).

2.1. The Life Prediction Calculations in Short Crack Growth Process (or Called the First Stage)

The life curves of short crack growth in crack forming stage (the first stage) are just described with curves 1 ( ), 2 and 3 ) in reversed direction coordinate system as in attached figure1, where About their constituting relationship and meaning of each curve are explained in [1-3], here for them are only described with the Multiplication method of two parameters in short crack growth process.

(1) Under work tress condition (high cycle loading)

In attached fig.1, under work tress condition, the life prediction equation corresponded to reversed curves 1 and 3 can be calculated in short growth process as following form

      (1-1)

or

   (1-2)

Where the  is defined as two-parameter stress-strain factor under monotonous loading; the  is defined as two-parameter stress-strain factor range under fatigue loading.

(2)

(3)

(4)

(5)

Here the eqn (2) is driving force of short crack growth under monotonic loading, and the eqn (3) is driving force under fatigue loading. . The in eqn (4) is defined the comprehensive material constant, that is corresponding reversed curves 1 () (attached fig.1), its mean stress ; The in eqn (5) is corresponding reversed curves 3() (attached Fig.1), its mean stress is . Here for mean stress  to adopt as the correctional method in reference [11]. Author research and think, the is a calculable parameter of having function relation with other parameters  , its unit ofis the "", its physical meaning is a concept of power, that is to give out an energy to resist outside force, it just is a maximal increment value to give out energy in one cycle, before the specimen material makes failure. Its geometrical meaning is a maximal micro-trapezium area approximating to beeline (Fig1), that is a projection of corresponding to curve 1 () or 3  on the y-axis, also is an intercept between. Its slope of micro-trapezium bevel edge just is corresponding to the exponent of the formula (4-5). Here the parametersandare respectively material constants under high cycle or low cycle fatigue. The,  is the fatigue strength exponent under high cycle fatigue; the,  is the fatigue ductility exponent under low cycle fatigue.

Here

(6)

Or

(7)

The  in eqn (6-7) is defined as an effective rate correction factor in first stage, its physical meaning is the effective rate to cause whole failure of specimen material in a cycle, its unit is the .  is a reduction of area.  is pre-micro size which has no effect for fatigue damage under prior cycle loading [12].  is an initial micro size, is a critical crack size before failure, is initial life,; is failure life,. It should yet point the  in eqn (1) is a transitional crack size transited from micro to macro damage, , is a macro crack value corresponded to forming macro-crack size .is a medial crack value between initial micro crack and transitional crack size corresponding medial life . By the way, here is also to adopt those material constants  as "genes" inside the fatigue damage subject. So, for the eqn (1), its final expansion equation corresponded reversed to curves 1  is as below form:

(8)

And its final expansion equation corresponded reversed to curves 3should be:

(9)

(2) Under work stress condition (or low cycle loading)

When under condition, due to plastic strain occurring cyclic hysteresis loop effect, its life equation corresponded to reversed direction curve is as following

       (10-1)

Or

    (10-2)

Therefore its final expansion equation for eqn (10) is as below form,

(11)

(12)

Here influence of mean stress in eqn (12) can be ignored. But it must point that the total strain range  in eqn (11-12) should be calculated by Masing law as following eqn. [13]

(13)

2.2. The Life Predicting Calculations in Long Crack Growth Process (or Called the Second Stage)

In Fig.1, the residual life curves of long crack growth are just described with curves 1' ( ), 2' and 3' ) in reversed direction coordinate system. For the driving force and the life calculating problems in long crack growth process, here to adopt the -method of two parameters and the stress- method are described and calculated for them.

(1) Under work tress condition

1) -factor method

For its life prediction equation corresponded reversed curves  and in fig.1 should be as below

(14)

Or

(15)

Where

(16)

(17)

Where the  is defined as two-parameter stress-strain factor under monotonous loading; the  is defined as two-parameter stress-strain factor range under fatigue loading, they are all drive force for long crack growth in second stage. Theis correction factor [14] related to long crack form and structure size. When corresponding curve (fig.1) its calculable comprehensive constant is as following

(18)

And the  corresponded to curve (fig.1), it should be as following form

(19)

Where theis a critical stress intensity factor, is mean stress intensity factor;  is a critical crack tip open displacement. It should be point that their physical and geometrical meanings for the  are all similar to that concept in short crack growth process mentioned above. But should explain the unit of  is .

Where

(20)

The  is defined to be the virtual rate, its physical meaning is an equivalent crack rate, that is the contributed crack growth rate in a cycle related to the precrack, the dimension can take similar to - dimension in reference [15-16], but the unit is different, because where its unit is , here it is the . The crack is a virtual crack size,  is an initial size as equivalent to a precrack. is an initial life, . is a virtual life, .

Moreover,

(21)

(22)

(23)

So the effective life expanded equation corresponding reversed direction curve is following forming.

(24)

In reference [17-18] refer to the effective stress intensity factor, same, here there are also two effective values and corresponding to the critical and , here to propose as follow

 or     (25-1)

           (25-2)

Where the  is a threshold stress intensity factor value. Thein (24) is an effective long crack size, it is obtained and calculated from eqns (21), (23) and (25-1), (25-2), and to take less value.

Such, the effective life expanded equation corresponding reversed direction curve should be

(26)

Its medial life in second stage is

(27)

2) -stress method

Due to word stress is still ( ), the long crack growth residual life equation of corresponding reversed direction curve in fig.1 is as following form

(28)

Here

(29)

(30)

The  -factor is a driving force model for long crack growth under monotonous load; the - factor is driving force under fatigue loading, which for above - factor and - factor are all calculated by the stress. And the comprehensive material constants  in eqn (28) can also expressed by the stress and to be also calculable material constant, that are as following:

(31)

(32)

So for , its final expansion equation corresponded to reversed direction curve  is as below form,

(33)

For, the life equation corresponded to reversed direction curve is following

(34)

(2) Under work tress condition

2) -factor method

Under condition, due to the materials occur plastic strain, the exponent of its equation also to show change from  to ; and due to occur cyclic hysteresis loop effect, its effective life calculable models corresponded to reversed curve  in attached figure 1 is as below form

(35)

Where  is also calculable comprehensive material constant, on exponent as compared with above eqn (18) and (19) that is not different,

(36)

(37)

Where  is an linear elastic exponent in long crack growth process, . And  is a ductility exponent, . It should note its unit of the is .

So the effective life expanded equation corresponded to reversed direction curve (attached fig.1) should be

For

(38)

For

(39)

Here

(40)

(41)

-stress method

Due to work stress , if adopt stress to express it, the  in eqn (35) should be

For

(42)

For

(43)

Therefore the residual life equation of corresponded to reversed direction curve in fig.1, its final expansion equation is as below form,

For ,

(44)

For ,

(45)

Here, influence to mean stress in eqn (45) usually can ignore. And it must be point that the units in the life equations are all cycle number.

2.3. The Life Prediction Calculations for Whole Crack Growth Process

Due to the behaviors of the short crack and the long crack are different, for availing to life calculation in whole process, author proposes it need to take a crack transitional size at transition point between two stages from short crack to long crack growth process, and the crack size at transition point can be derived to make equal between the crack growth rate equations by two stages, for instance [19],

(46)

Here the equation (46) is defined as the crack-growth-rate-linking-equation in whole process.

(1) Under work stress

For, its expanded crack-growth-rate-linking-equation for (46) corresponding to positive curve  (attached fig.1) is as following form

(47)

For, its extended crack-growth-rate-linking-equation for (45) corresponding to positive curve  (attached fig.1)is as following form

(48)

And the life equations in whole process corresponding to reversed direction curves and  should be as below

(49)

Its expanded equation corresponding to reversed direction curves  is as following form

(50)

But for , its expanded equation corresponding to reversed direction curves  should be

(51)

(2) Under work stress

Under work stress, its expanded link rate equation for eqn (46) corresponded to positive curve  is as following form

(52)

And the life equations in whole process corresponding to reversed direction curve (attached fig.1)should be as following

(53)

So the expanded life prediction expression in whole process corresponded to reversed curve , it should be

(54)

It should point that the calculations for rate and life in whole process should be according to different stress level, to select appropriate calculable equation. Here must explain that its meaning of the eqns (46-48, 52) is to make link between the first stage rate and the second stage rate, Calculation method for which it should be calculated by the short crack growth rate equation before the transition point; it should be calculated by the long crack growth rate equation after the transition point , that is not been added together by the rates for two stages. But for the life calculations in whole process can be added together by two stages life. About calculation tool, it can be calculated by means of computer doing computing by different crack size [21].

3. Computing Example

3.1. Contents of Example Calculations

To suppose a pressure vessel is made with elastic-plastic steel 16MnR, its strength limit of material, yield limit, fatigue limit , reduction of area is , modulus of elasticity; Cyclic strength coefficient , strain-hardening exponent ; Fatigue strength coefficient, fatigue strength exponent , ; Fatigue ductility coefficient , fatigue ductility exponent , . Threshold value, critical stress intensity factor . Its working stress ,  in pressure vessel. Suppose that for long crack shape has been simplified via treatment become an equivalent through-crack, the correction coefficientof crack shapes and sizes equal 1. Other computing data are all in table 1.

Table 1. Computing data.

92.7 28.23 8.6

3.91 0.18 2.9 1.0

3.2. Required Calculating Data

Try to calculate respectively as following different data and depicting their life curves

(1) To calculate the transitional point crack sizebetween two stages;

(2) To calculate the crack growth rate for transitional point ;

(3) To calculate the life in first stage from short crack growth  growth to transitional point ;

(4) To calculate the effective life in second stage from transitional point crack size to long crack;

(5) Calculating the whole service life

(6) To depict life curves in whole process.

3.3. Calculating Processes and Methods

The concrete calculation methods and processes are as follows

(1) Calculations for relevant parameters

1) Total strain

To calculate stress-strain data by reference (31):

,

;

2) Stress range calculation:

3) Mean stress calculation:

4) Total strain range:

5) Elastic strain range:

;

6) Plastic strain range:

7) According to formulas (7), calculation for correction coefficient in first stage

8) Take virtual rate  in second stage by eqn (20)

  

9) According to formulas (38), Calculating effective size

 take .

Here take effective crack sizein first and the second stage.

(2) To calculate the transitional point crack sizebetween two stages:

1) Calculation for comprehensive material constant in first stage by eqn (5)

2) Calculation for comprehensive material constant in second stage by eqn (43)

.

3) Calculation for transitional point crack size

According to the equations (46) and (52),

Then, for the transitional point crack size between two stages it can make link between the rate expansion at left side and the rate one at right side as following

;

Then make simplified calculation:

;

So obtain the transitional point crack size between two stages.

The crack growth rate calculations for transitional point

Here it can be seen, the crack growth rate at the transition point crack size  is same, it is .

(3) Life prediction calculations in whole process

1) Select life predicting calculation equation (12), the calculable life in first stage from short crack to transitional point  is as follow,

So first stage life

And from above we can derive simplified life equation in first stage corresponded to different damage value as follow form  

Select life predicting calculation equation (45), the calculable life in second stage from transitional point crack size  to  is as follow,

From above formula we can also derive simplified life equation corresponding different damage value as follow form à

Therefore, whole process life is  

If use integral equation (53) and (54) to calculate the service life in whole process, it is

3.4. Calculating Results

3.4.1. Life Data of Each Stage and Whole Process

Calculating results are that the first stage life is  cycle from micro-crack size 0.02 mm to transition point size 1.113mm; the second stage life is  cycle from transition point-crack size 1.113 mm growth to long crack length 5 mm; and the service life in whole process is  cycle. This result is consistent with expansion equations calculation results data .

The life data corresponded to different crack length are all included in table 2, 3 and 4. Author finds these data are calculated by two-parameter-multiplication-method that are basically close as compared with another result data: which are calculated by the single-calculated method, it has been published recently in reference [21]. It should be point that the data calculated by two-parameter-multiplication-method are nicer and steady in whole process.

Table 2. Crack growth life data in whole process.

Data point of number 1 2 3 4 5
Crack size (mm) 0.02 0.04 0.1 0.2 0.4
Data of the first stage 15527950 7763975 3105590 1552795 776398
Data of the second stage Useless section    

Table 3. Crack growth life data in whole process.

Data point of number 6 7 8 transition point 9
Crack size (mm) 0.6 0.7 1.113 1.5
Data of the first stage 517598 44365 274103 207039
Data of the second stage 1341644 912931 274240 136076

Table 4. Crack growth life data in whole process.

Data point of number 9 10 11 12 13
Crack size (mm) 1.5 2.0 3.0 4 5
Data of the first stage 207039 155280 Useless section
Data of the second stage 136076 66336 24097 11747 6728

3.4.2. To Depict the Life Curves in Each Stage and Whole Process

According to crack growth data for deferent crack size depicting the curves in whole process, which are showed in figures 2 and 3.

Figure 2. Life curve in whole course (in decimal coordinate system).

(A) 2-1-data curve in first stage obtained by two-parameters calculating method;

(B) 2-2-data curve in second stage obtained by two-parameters calculating method;

(C) This example transition point from micro-crack size 0.02mm to long crack size 5 mm is just at eighth point (crack size ).

Figure 3. Life curve in whole course (in logarithmic coordinate system).

(A) 2-1-data curve in first stage obtained by two-parameters calculating method;

(B) 2-2-data curve in second stage obtained by two-parameters calculating method;

(C) This example transition point from micro-crack size 0.02mm to long crack size 5mm is just at eighth point (crack size).

4. Discussions

Author puts forward such a view point: in the mechanics, the aviation, the machinery and the civil engineering etc fields, there are a scientific law of similar to gene principle and cloning technique. Because where there are common scientific laws as life science:

1) Each unit cell combined in a genetic structure has all its own genetic (or inheritable) character;

2) To have the clonable (or can copy) and the transferable characters, and can be recombined together by inherent relationship;

3) The new combined structures there are also with new characters and functions.

Please note: Those parameters inside from equations (1) to (54) in this paper, which there are all as "genetic character of unit cells", e.g. as the stress parameter, the strain and their material constantsetc subjected traditional materials mechanics; and these "genetic elements" all to maintain their original calculable properties and functions. But once they are transferred into new areas as the fracture mechanics or the damage mechanics, they have been formed by new structure-equations. Then the new equations have also been shown with new calculable functions, as those equations in the traditional material mechanics. Author just is according to such thinking logics and methodologies, and by means of various relatedness among the material constants, e.g. ;  etc, and according to the cognition for their physical and geometrical significance for key parameters, thereby derives the above mentioned a lot of calculable models.

5. Conclusions

In above text, author adopts the -method and the -factor method to calculate various parameters and a mass of data, here can obtain following common conclusions:

(1) About comparison of calculating results: As compared with the single-parameter method to find, that above mentioned calculated data by the multiplication-method of two-parameters are nicer and steady in whole process.

(2) About new cognitions for key parameters: True material constants must show the inherent characters of materials, such as the  and  etc in the material mechanics; for instance the, , ,  and so on in the fatigue- damage mechanics; for example the, and and so on in the fracture mechanics; which could all be checked and obtained from general handbooks; But some new material constants in the fracture mechanics that are essentially to have functional relations with other materials constants, for which they can be calculated by means of the relational conventional materials constants under the condition of combining experiments to verify, e.g. eqns (4-5), (18-19), (31-32), (36-37), (42-43), (46-48), (52), etc. Therefore for this kind of material constants should be defined as the comprehensive materials constants.

(3) About cognitions of the physical and the mathematical meanings for key parameters: The parameters  in the first stage and in the second stage, Their physical meanings are all a concept of the power, just are a maximal increment value paying energy in one cycle before to cause failure. Their geometrical meanings are all a maximal micro-trapezium area approximating to beeline.

(4) About the methods for crack propagation rate and lifetime calculations: Calculation for crack transition size it can be calculated form the crack-growth-rate-linking-equations (46-48) and (52) in whole process; before the transition point it should be calculated by short crack growth rate equations ; after transition point it should be calculated via long crack growth rateequations. But for the lifetime calculations in whole process can be added together by life cycle number of two stages.

(5) Based on the traditional material mechanics is a calculable subject; in consideration of the conventional material constants there are "the hereditary characters"; In view of the relatedness and the transferability between related parameters among each disciplines; And based on above viewpoints and cognitions of the (1)~(4); then for the fatigue and the fracture disciplines, if make them become calculable subjects, that will be to exist possibility.

Acknowledgments

At first author sincerely thanks scientists David Broek, Miner, P. C. Paris, Coffin, Manson, Basquin, Y. Murakami, S. Ya. Yaliema, Morrow J D, Chuntu Liu, Shaobian Zhao, Jiazhen Fan, etc, they have be included or no included in this paper reference, for they have all made out valuable contributions for the fatigue-damage-fracture subjects. Due to they hard research, make to discover the fatigue damage and crack behavioral law for materials, to form the modern fatigue-damage-fracture mechanics; due to they work like a horse, make to develop the fatigue-damage-fracture mechanics subjects, gain huge benefits for accident analysis, safety design and operation for which are mechanical equipments in engineering fields. Particularly should explain that author cannot have so many of discovery and provide above the calculable mathematical models and the combined figure 1, if have no their research results.

Author thanks sincerity the Zhejiang Guangxin New Technology Application Academy of Electromechanical and Chemical Engineering gives to support and provides research funds.

Nomenclature

and

initial crack and medial short crack in the first stage

and

crack transition size from short crack to long crack and macro crack size,

medial life in first stage,

life in first stage

,

fatigue strength exponent under high cycle fatigue in first stage,

,

fatigue ductility exponent under low cycle fatigue in first stage,

comprehensive material constant in first stage

reduction of area

fatigue strength coefficient under fatigue loading

fatigue ductility coefficient under fatigue loading

and

two-parameter stress-strain factor and stress-strain factor range of short crack in first stage

initial crack size in the second stage

critical crack size to make fracture in one cycle

,

fatigue strength exponent under high cycle fatigue in second stage,

,

fatigue ductility exponent under low cycle fatigue in second stage,

comprehensive material constant in the second stage

effective comprehensive material constant in the second stage

crack tip open displacement and crack tip open displacement range in second stage

effective stress intensity factor in second stage

effective crack tip open displacement in second stage

mean stress intensity factor

critical crack tip open displacement

fracture toughness.

,

two-parameter stress-strain factor and stress-strain factor range of long crack in second stage

virtual rate in the second stage

initial, medial , effective and critical size of crack in the second stage,

initial life, medial life , effective and critical life in second stage,

life in second stage

lifetime in whole process

References

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