Cyclic life prediction of a specimen underrotational bending test

Chethan Mohan Kumar950843

January 12, 2012

Under the guidance of :

M.Sc.Dipl.-Ing. Henry Pusch

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Abstract

Fatigue testing of a material holds a distinct position in the agendaof testing materials since in real time applications. More often it is byfatigue failure a material fails. Any product designed is intended tofunction for a certain number of cycles. It must be ensured that thematerial does not fail before its intended life since failure by fatiguereflects no prior indications .Hence it is imperative that accurate dataof cyclic life with respect to various stress levels must be tabulatedfrom experiments or by predictions. An attempt to predict the cycliclife of materials using the two approaches of stress life methods andthe strain life methods is the kernel of this project.The focus here ismainly on a specimen subjected to the Rotational Bending machine.This implies that the conditions considered for theoritical predicitionswould also imbibe similar loaded states. A shoulder specimen witha particular radius is considered. The accuracy of results obtainedtheoretically is then compared with experimental data.The Governingequation for stress life method is the Basquin’s equation and for strainlife method is the Mason-Coffin relation.The material used is predom-inantly AISI 1045 and occasionally AISI-1035.A function in scilab forboth the equations has been coded for the ease of calculations.

Keywords:Fatigue, Basquin’s equation, Mason-Coffin relationship,Rotational Bending machine, Woehler curve

Notations:M : Bending momentI : Surface moment of inertiay : Distance between center of the specimen and a generic pointS : Cyclic stress,MPaSe : Endurance limit, MPaA : Multiplication factor in Basquin’s equationb : Exponent or slope in Basquin’s equationSe

′: Modified endurance limit,MPa

Sut : Ultimate tensile strength, MPaSut(ck-45) : 620 MPaδε2 : Strain amplitude, mm

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Acknowledgements

I sincerely thank M.Sc.Dipl.-Ing. Henry Pusch for giving me an opportunityto work on this project which has helped me to comprehend the theoreticalaspects of fatigue practically by conducting experiments in the laboratory.Iwould also like to thank him for his constant guidance and consistent supportin accomplishing the end points defined in the project successfully.

I am also grateful to my colleague Mr. Daniel Korner who skillfully pro-duced the specimens required for conducting the experiments. The quality ofspecimen produced by him is quite commendable. I would also like to thankmy friend Mr. Narendra Komarla who accompanied and co-ordinated withme in conducting the experiments.

I finally like to thank The Department of Material Science and Universityof Wuppertal for providing all the required facilities.

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Contents

1 Introduction 51.1 Fatigue failure . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.2 Fatigue life methods . . . . . . . . . . . . . . . . . . . . . . . 6

2 Rotational Bending Machine 82.1 Specimen Geometry . . . . . . . . . . . . . . . . . . . . . . . . 82.2 Loading conditions . . . . . . . . . . . . . . . . . . . . . . . . 8

3 Stress life method 93.1 Woehler Curve . . . . . . . . . . . . . . . . . . . . . . . . . . 103.2 Basquin’s Equation . . . . . . . . . . . . . . . . . . . . . . . . 113.3 Example Calculation . . . . . . . . . . . . . . . . . . . . . . . 12

4 Strain life method 154.1 Mason-Coffin method . . . . . . . . . . . . . . . . . . . . . . . 16

5 Evaluation of cyclic life with Scilab 185.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185.2 Function for Basquin’s equation . . . . . . . . . . . . . . . . . 195.3 Function for Mason-Coffin equation . . . . . . . . . . . . . . . 21

6 Conclusions 23

Appendices 25

A Rotational Bending machine 25

B Data for Basquin’s Equation 26

C Data for Mason-Coffin relationship 28

D Kf Table 29

E Experimental Data 30

F Scilab functions 30F.1 cyclicstress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30F.2 cyclicstrain . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

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1 Introduction

Fatigue is a phenomenon occuring on a regular basis when a load is appliedon a particular object and removed cyclically. We come across a series ofdiurnal products that we use which undergo the fatigue phenomenon. Thechair which we sit on nuemerous times undergoes fatigue and is destined tofunction for a certain number of cycles. The same case holds with the closingand opening of a laptop. Practically, if tensile strength is a material property,fatigue strength can be correlated to durability of the material.

Fatigue occurs at a stress level lesser than the tensile strength of thematerial. But this stress level is applied over a series of loading and unloadingfor a number of cycles. We can comprehend that the stresses accumulate overa period of time which confronts us with the concept of hysteresis loop.

Hysteresis refers to systems that may exhibit path dependance, or ”rateindependent memory”[1].In fatigue, Hysteresis loops play a significant role indetermining whether the material undergoes material hardening or materialsoftening.

Figure 1: Hysteresis loop with material hardening

Material hardening occurs if the stress required to propagate the crackfurther increases as the number of cycles of tension and compression on thematerial increases.As we observe in Figure 1, the stress required to increasestrain is higher than the required stress in the previous cycle. Materialsoftening is the phenomenon where the stress required for crack propagationdiminishes with the number of cycles the material undergoes. These two phe-nomenon influences the way a material behaves when subjected to a fatigue

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test. Though we are dealing with the rotational bending test, the hysteresisloop from the tension-compression test gives us an insight into the processthat may occur in the specimen when it undergoes fatigue.

1.1 Fatigue failure

The attributes of failure by fatigue will be discussed in this section. A fa-tigue failure has an appearance similar to a brittle fracture, as the fracturesurfaces are flat and perpendicular to the stress axis with the absence of neck-ing[2].The features and occurence of fatigue failure is different from brittlefracture. Fatigue failure occurs in three stages which are the crack initiationphase, Crack propagation phase and fracture.

1. Crack Initiation: As the name suggests, an initial crack is formed inthis phase. It may be formed due to the existence of certain irregular-ities in the specimen.The crack formed can be termed as microcrackssince they are not visible to the unaided eye. For example: Presence ofnotches, a radius at a certain position which is design driven, or evenscratches or surface irregularities in the specimen. These irregularitiesact as stress raisers increasing the localized stresses. Thus a crack isformed as a result of their existence. In this project, the stress raiseris due to a radius in the specimen used in the rotational bending ma-chine which will be dealt in explicit detail under the section ”Rotationalbending machine”.

2. Crack Propagation: The microcracks propagate and convolute intolarger or macrocracks.These macrocracks form parallel plateau-like frac-ture surfaces separated by longitudinal ridges.During cyclic loading,these cracked surfaces open and close, rubbing together, and the beachmark appearance depends on the changes in the level or frequency ofloading and the corrosive nature of the environment[2].

3. Fracture: In this stage, the material fails which is the last cycle ofstress it undergoes. This is the result of the remaining material unableto support the loads leading to failure. The fracture may be brittle,ductile or a combination of both.

1.2 Fatigue life methods

There are three main approaches of fatigue life methods which are Stress lifemethod , Strain life method and fracture mechanics method. The terms low

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Figure 2: Fracture surface of a specimen(Ck-35/AISI-1035 steel) tested onrotational bending machine

cycle fatigue and high cycle fatigue has to be dealt before we proceed anyfurther. Number of cycles ranging from 1 ≤ N ≤ 1000 is considered as lowcycle fatigue where as N > 1000 cycles is considered as high cycle fatigue.The distinction contributes to the predictions in the fatigue life methods.

1. Stress life method is a classical approach of cyclic life predictionbased on the stress levels only. This is considered as the least accuratemethod especially in the region of low cycle fatigue. But it is easilydeciphered and uncomplicated since the data required for the calcula-tions are not too many. It can also be implemented to a wide range ofdesign applications and represents the high cyclic fatigue accurately.

2. Strain life method method involves more detailed analysis of theplastic deformation at localized regions where the stresses and strainsare considered for life estimates[2]. It is more complex since it involvesmany parameters to be considered. The Strain life approach generatesaccurate results in the low cycle fatigue range.

3. Fracture mechanics method holds good under an assumption thata crack is already fromed in the specimen on account of the presence ofirregularities. This method is more suitable for practical applications.

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In this project, we consider the Stress life approach and the Strain life ap-proach to predict the cyclic life of specimens used for the rotational bendingmachine.

2 Rotational Bending Machine

In the rotating bending machine, a rotating sample is loaded with a singleforce. The load is applied at one end of the sample and with the help ofa motor, rotation about its own axis is achieved. Due to this rotation , aload reversal condition is achieved at two opposite sides on the circumferenceof the specimen.A triangular bending moment is developed in the specimen.Bending stress can be calculated from the equation

σ(y) =M

I.y (1)

Since failure of the specimen occurs from the weakest point situated in prox-imity of the surface, the test employs an ideal approach for testing.

The salient features of the rotational bending machine is the demonstra-tion of fatigue resistance under rotational bending load, the S-N diagram orwoehler curve can be recorded with varying the load applied, investigationof different specimen geometries i.e the influence of notches or the influenceof surface finish can be measured.

2.1 Specimen Geometry

The specimen used for the test is a shoulder specimen with a radius at theshoulder. It has been designed in such a way for experimental purposes todefine the position of failure since the von mises stress is maximum at thisregion. Further the radius also gives a variation for testing possibilities sincethe effect of varying radii on fatigue can also be determined from the machine.An overview of the specimen tested is shown in the figure.

2.2 Loading conditions

It is important to analyse the conditions undergone by the sample during therotational bending test in order to apply it for the calculations involved incyclic life time predictions. Since the loading here is reversed along the axis,the influencing factor of mean stress can be considered to be zero and theratio of maximum stress to minimum stress,stress ratio is thus defined as

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Figure 3: Specimen used in Rotational bending machine

Figure 4: Loading configuration

R =σmaxσmin

= −1 (2)

which implies that the corressponding models which do not consider meanstress effects must be used for calculations.An overview of loading is shownin figure 4.

A more circumspective look over the experimental set up would raise thequestion of whether rate of loading affects the fatigue life of specimen sincerotation is achieved by speed of the motor. Though rate of loading is aninfluencing factor, it does not apply in this case since it is dominant only incertain materials which excludes steel which we have considered.

3 Stress life method

Stress life method is a classical way of predicting cyclic life of materials.As the name suggests, it subjects the specimen under varying stress andthe number of cycles of stress reversal is counted till the specimen fails.The starting point of stress is at a stress level below the ultimate tensile

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strength of the material. Subsequently the stress applied is reduced and thecorressponding number of cycles are tabulated.An S-N curve or a Woehlercurve is plotted from these tabulations. At a certain stress level it is observedfor certain materials that the curve becomes horizontal with the increase ofstress level. This phase is termed as infinite life of a specimen which means itnever fails. This infinite life is attained at 106 cycles or 107 cycles dependingon the type of material.

Historically, most attention has been focussed on situations that requiremore than 104 cycles where stress is low and primarily elastic[1]. This ischaracteristic of stress approach since it does not explicitly calculate theplastic deformations occuring in the specimen during fatigue. Hence thestress approach is generally more accurate during high cycle fatigue but notrecommended for low cycle fatigue because of the presence of plastic defor-mations.

3.1 Woehler Curve

An S-N curve for a material defines alternating stress values versus the num-ber of cycles required to cause failure at a given stress ratio.As mentionedearlier, we have a stress ratio of −1 for our experiments. A typical S-N curveis shown in the figure. The Y- axis represents the alternating stress (S) andthe X-axis represents the number of cycles (N). An S-N curve is based on astress ratio or mean stress. You can define multiple S-N curves with differentstress ratios for a material. The software uses linear interpolation to extractdata when you define multiple S-N curves for a material.

S-N curves are based on mean fatigue life or a given probability of failure.Generating an S-N curve for a material requires many tests to statisticallyvary the alternating stress, mean stress (or stress ratio), and count the num-ber of cycles.Material performance is commonly characterized by an S-Ncurve or Woehler curve in high-fatigue region.

The results are plotted on semilog paper or a log log paper. In somecases the graph becomes horizontal after the material has been stressed fora certain number of cycles. Plotting on log paper emphasizes the bend inthe curve which might not be apparent if the results are plotted by usingCartesian coordinates.

In most of the cases an S-N curve may be found to be a straight line.Thisis the result of a log-log scale in both ordinate and axis. The importanceof fatigue in the high cycle region is generally emphasized with the startingpoint of the curve(line) at 103 cycles but not at 100 cycles and the endpointat infinite life of the material.

The probabilistic nature of fatigue emphasizes the consideration of various

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Figure 5: S-N curve showing different fatigue regions.1-Time dependant fa-tigue resistance 2-Fatigue rupture 3-Absence of fatigue fracture

factors which interfere in the accurate prediction of cyclic life. Nevertheless,correction factors based on geometry, surface roughness make sure that thisanomality is fixed.

3.2 Basquin’s Equation

Basquin’s equation is a stress life method of predicting cyclic life. The gov-erning equation can be written as

S = A(N)b (3)

This leaves us with the calculation of two factors, the multiplication factorA and exponent or slope b. The cycles N is dependant on the stress level S.

As mentioned earlier, S-N curves are in some cases represented as a linecan be clearly demonstrated using this equation when a log-log plot of S-Nis drawn. Consider the governing equation

S = A(N)b (4)

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When log of this equation is considered,

log(S) = log(A) + b log(N) (5)

which clearly replicates the equation of a line

y = c+mx (6)

Hence in order to generate the curve from the basquin’s equation, the numberof boundary conditions required is exactly the number of boundary conditionswe require to draw a line which are two points. Thus we need to know twopoints in the S-N curve in order to plot the Basquin curve.

It is logical to consider these points depending on material characteristicssince it implies that it remains unchanged if a specific material is used. Oneof the points considered is the ultimate strength of the material for one cycle(Ultimate tensile strength,1) and the other point considered is the endurancestrength Se of a material, which is the stress level where infinite life beginsand the line becomes horizontal. Endurance strength is generally between106 or 107 depending on certain materials. Thus we have our second point(Se,106 or107).

From the two points obtained, we generate the slope of the basquin line,b and intercept log(A). Thus we have the governing Basquin’s equation togenerate the S-N curve.

Sometimes the starting point of the line is considered at the beginingof high cycle fatigue, 103 cycles with the stress level of 0.9∗Ultimate tensilestrength.The starting point thus becomes (0.9∗Ultimate tensile strength,1000).This is to make sure that the high cycle fatigue region curve is accuratelyfollowed with the original behaviour by the equation. A better picture of thewhole calculation process is obtained in the following subsection.

3.3 Example Calculation

Consider the specimen used in the rotational bending machine from Figure 3.The material used is Ck-35/AISI 1035 steel. The tensile strength is 560MPa.

The governing Basquin’s equation is

S = A(N)b (7)

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1. Determining the two points:

To Calculate the stress levels at the two points of cyclic life. The firststress level at the begining of high cycle fatigue, 1000 cycles and thesecpond point at infinite life, at 2000000 cycles.

2. First point for the line

The stress level at 1000 cycles is given by

S1000 = 0.9xUltimate tensile strength(8)

3. Second point for the line

The second point considered is (Se, 2000000). To have a meaningfulvalue for Se, we need to consider certain factors. Generally

Se = 0.5xUltimate tensile strength(9)

But from Figure 3, it is evident that there exists a shoulder in the geom-etry of the specimen. This induces a stress concentration in the shoul-der specimen used for the rotational bending machine.This stress con-centration reduces the endurance limit of the specimen further. Hencethe modified Endurance limit, Se

′must be calculated to generate the

second point required for the line.

4. Calculation of Se′:

Se′=

SeKf

(10)

5. To calculate Kf :

Kf is the fatigue stress concentration factor generally defined as a re-duced stress concentration factor from Kt , the stress concentrationfactor beacuse of lessened sensitivity of notches.

The relation of Kf , Kt and notch sensitivity q is given by

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q =Kf − 1

Kt − 1(11)

In our calculation, we find Kt first from the geometry of the specimen.Thus Kt for our specimen with D/d ratio of 1.5 and r/d ratio of 0.25from the chart is found to be 1.35.

There are two ways of finding notch sensitivity q. One from the notchsensitivity chart and the other by the notch sensitivity equation,

q =1

1 +√a√r

(12)

√a is called the Heywood’s parameter and can be obtained from the

data table. It is found that for our specimen , Heywood’s parameter is139/Sut which results in q from equation 12 to be 0.85.

Now that we have

q = 0.85 Kt = 1.28

We can find Kf from equation 11 and get,

Kf = 1.238

Subsequently substituting this value in equation 10 we get,

Se′ = 226.2

6. Final Basquin’s equation:

Now that we have the two points,

(S1000, 1000) = (504, 1000)

(Se′, 2000000) = (226.2, 2000000)

Substituting and solving in the Basquin’s equation,

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log 504 = logA+ b log(1000)

log 226.2 = logA+ b log 2000000

Subtracting the above equations, we obtain the value of b.

b =log 504− log 226.2

log 1000− log 2000000= −0.1054

By substituting b in one of the equations, we obtain

A = 1043.85

Thus the equation to generate the curve is written as,

S = 1043.85(N)−0.1054

4 Strain life method

Though strain life method is considered here in order to show an alter-native way to calculate the cyclic life of a material, it is an indegenousmethod since it calculates the cyclic life considering the plastic defor-mations occuring at higher stress levels in the material. This places thestrain life method as one of the accurate ways of predicting the cycliclife in low cycle fatigue regions.

A fatigue failure almost begins at a local discontinuity. When thestress at this discontinuity exceeds the elastic limit, plastic strain oc-curs. Test specimen subjected to reversed bending are not suitable forstrain-cycling because of the difficulty to measure plastic strains. Hencewe cannot compare the experimental results and the theoretical calcu-lations. Nevertheless , the calculation process using the Mason-Coffinmethod will be briefly demonstrated in the following section.

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Figure 6: True stress-strain hysteresis loops showing the first five stress re-versals of a cyclic-softening material

4.1 Mason-Coffin method

The Mason-Coffin relationship considers various factors in order to re-late the strain amplitude and cyclic life.

(a) Fatigue ductility coefficient, εf′ is the true strain corressponding

to fracture in one reversal. Point A in Figure 6. The plastic strainline begins at this point in Figure 7.

(b) Fatigue strength coefficient, σf′ is the true stress corresponding to

fracture in one reversal (point A in Figure 6). The elastic strainline begins at σf

′/E in Figure 7.

(c) Fatigue ductility exponent c is the slope of the plastic strain linein Fig 7 and is the power to which the life 2N must be raised tobe porportional to true plastic strain amplitude. Note that 2N isthe number of stress reversals and N is the number of cycles.

(d) Fatigue strength component b is the slope of the elastic strainline , and is the power to which the life 2N must be raised to beproportional to the true stress amplitude.

From Figur 6 , the total strain is the sum of elastic and plastic straincomponents. Therefore the total strain amplitude is

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Figure 7: A log-log plot showing how the fatigue life is related to the true-strain amplitude for hot rolled steel

δε

2=δεe2

+δεp2

(13)

The equation of the plastic strain line in Fig 7 is

δεp2

= δεf (2N)c (14)

The equation of the elastic straight line is

δεe2

=σF′

E(2N)b (15)

Hence from equation 13 , we have the total strain amplitude,

δε

2=σF′

E(2N)b + δεf (2N)c (16)

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The above equation is the Mason-Coffin relationship.

The parameters of the equation σF′, E, b, εf , c are obtained from the

data table.It can clearly be seen that the relationship between the strainamplitude and number of cycles(N) is non linear.Hence there involvesa comlexity in solving these kind of non-linear equations analytically.

But since we know the range of the number of cycles we are interestedin, normally from 100 cycles to 107 cycles, it is practical to loop thisrange of N in a program and find the domain. And just run the loopfrom N=100 to 107 cycles and find the strain amplitude which resultsin accurate solution. The program used in this case is Sci-lab which isdiscussed in the following section.

5 Evaluation of cyclic life with Scilab

5.1 Introduction

Scilab is a software for numerical mathematics and scientific visualiza-tion. It is capable of interactive calculations as well as automation ofcomputations through programming. It provides all basic operationson matrices through built-in functions so that the trouble of develop-ing and testing code for basic operations are completely avoided. Itsability to plot 2D and 3D graphs helps in visualizing the data we workwith. It is quite similar to Matlab and a Matlab-Scilab translator isalso embedded in the software. The advantage with scilab is that it isavailable an open source and students can be benifited from this.

We use the software here in order to perform computation to calculatethe cyclic life from the governing equations and to plot a Woehler curveor a Strain versus Life curve. Even though the program is not socomplicated , it is effective with the rigorous computations and sparesus the time and effort.

In this project, two functions have been implemented. One func-tion which generates the Woehler curve cyclicstress(Ultimate ten-sile strength, Kf ) and the other function to generate the Strain-lifecurve cyclicstrain(fatigue ductility coeff, fatigue strength coeff, c ,b,Young’s).

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5.2 Function for Basquin’s equation

The template for the Basquin equation is

cyclicstress(Ultimate tensile strength, Kf )

The first parameter is characteristic of a material and the second pa-rameter must be calculated and then fed to the function. The com-plexity in integrating the calculation of Kf in the function is that itinvolves reading values from a chart. Hence it is imperative to calculateKf as shown under the Example calculation subsection. To substitutethis, a table containing Kf values for various cases of Shoulder radius(r), Larger diameter(D) and smaller diameter (d) can be found in theappendix.

Figure 8: Example for input to cyclicstress function

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Figure 9: Woehler curve for the input

An example of an input and the resulting Woehler curve is shown inthe two figures.

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5.3 Function for Mason-Coffin equation

The template for Mason-coffin equation is

cyclicstrain(fatigue ductility coeff, fatigue strength coeff, Fatigueductility exponent , Fatigue strength component ,Young’s Moadulus)

The five parameters to be inserted is obtained from the data table. Thedata table can be found under Appendix. An example for an input andthe resulting Strain-life curve are shown.

Figure 10: Example for input to cyclicstrain function

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Figure 11: Strain-life curve for the input

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6 Conclusions

The predicament in the analytical calculation of Fatigue life is thatit can only be an approximation since Fatigue is more statistical innature. This implies that, the experimental procedure involved mustconsider more number of trials for a particular stress level in order toconverge to a steady value of cyclic life.

The experimental and analytical values are compared for a ck-45 spec-imen. The ultimate tensile strength of ck-45 is taken as 620 MPa.The results of both the approaches are plotted.

In this project, the surface roughness of the specimen has not beenaccounted for. This can be one of the influencing factors in the results.Nevertheless, when the trend line of the experimental values is plottedusing a power law, the Basquin’s approach considered in this project isaccurate with the experimental data obtained from the rotating bend-ing machine which is clearly seen in the figure below.

Figure 12: Comparison of the plots of Experimental values vs. Analyticalvalues

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Figure 13: Comparison of the plots of Experimental values vs. Analyticalvalues

Numerically, the analytical and Experimental values are notably differ-ent. This is because the calculation involves logarithmic function andeven a slight variance in the value results in large differences. Though,this must not be the criteria for the comparison of accuracy betweenthe analytical and experimental methods, it must be noted that thetrend of the two curves using the power law is accurate.

It can also be inferred that the Stress method is more accurate atconsiderably lower stress levels in the High cycle fatigue region.

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Appendices

A Rotational Bending machine

Figure 14: Rotational Bending machine

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B Data for Basquin’s Equation

Figure 15: Heywood’s parameter√a for steels

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Figure 16: Stress concentration factors Kt for bending of a stepped bar ofcircular cross section with a shoulder fillet (based on photoelastic tests ofLeven and Hartman 1951; Wilson and White 1973)

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C Data for Mason-Coffin relationship

Figure 17: Data table for Mason-Coffin relationship

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D Kf Table

Figure 18: Data table for Kf values

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E Experimental Data

Figure 19: Experimental values for ck-45 steel on the Rotating Bendingmachine

F Scilab functions

F.1 cyclicstress

function c y c l i c s t r e s s ( u l t t e n s i l e , k f )S 1000 =0.9∗ u l t t e n s i l e ;S e =0.5∗ u l t t e n s i l e ;S e n=S e / kf ;m=log ( S 1000 )−log ( S e n ) ;n=log (1000) −log (10000000) ;b=m/n ;p=log ( S 1000 ) −b∗ log (1000) ;a=exp(p) ;t i t l e ( ”STRESS APPROACH” ) ;

x l a b e l ( ” l og (N) ” ) ;y l a b e l ( ”STRESS,MPa” )

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for s t r e s s=S e n : u l t t e n s i l eg=log ( s t r e s s )−log ( a ) ;h=g/b ;l i f e=exp(h) ;

disp ( s t r e s s , l i f e ) ;plot ( log10 ( l i f e ) , ( s t r e s s ) , ’ . ’ ) ;

end

endfunction

F.2 cyclicstrain

function c y c l i c s t r a i n ( fatduc , f a t s t r , c , b , youngmod)x=f a t s t r /youngmod ;

t i t l e ( ”STRAIN APPROACH” ) ;x l a b e l ( ” l og (N) ,Number o f c y c l e s ” ) ;y l a b e l ( ” l og (Ea) ,STRAIN AMPLITUDE in mm” ) ;

for j =1 :0 .01 :6l i f e =10ˆ j ;y=l i f e ˆb ;z=l i f e ˆc ;ea=x∗y+fatduc ∗z ;disp ( l i f e , ea ) ;plot ( log10 ( l i f e ) , log10 ( ea ) , ’ . ’ ) ;

endendfunction

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