material data derivation for the fatemi and socie …1021821/fulltext01.pdfimprove fatigue life...

MASTER’S THESIS

2007:027 CIV

KRISTINA OLAUSSON

Material Data Derivationfor the Fatemi and Socie

Critical Plane Model

MASTER OF SCIENCE PROGRAMMEEngineering Physics

Luleå University of TechnologyDepartment of Applied Physics and Mechanical Engineering

Division of Solid Mechanics

2007:027 CIV • ISSN: 1402 - 1617 • ISRN: LTU - EX - - 07/27 - - SE

ii

Preface This work has been carried out in a period from August 2006 to January 2007, at the department of Solid Mechanics and Structural Dynamics at Volvo Aero Corporation in Trollhättan. The thesis is the final part of the Master of Science Program in Engineering Physics at Luleå University of Technology and corresponds to twenty university credits. First, I would like to thank my supervisor Leif Samuelsson for all his help and support during this thesis. Also thanks to Anders Larsson for his help with the software problems and to Peter Georgsson for his help regarding material data. In addition, I would like to thank the people at division 7162 for a pleasant working environment. Finally, I would like to thank my examiner at Luleå University of Technology, Karl-Gustaf Sundin. Trollhättan, January 2007 Kristina Olausson

iii

Abstract During the last decades critical plane models have been developed as tools for crack initiation analyses within fatigue. Compared to older methods, these models aim to improve fatigue life predictions in multiaxial and non-proportional loading situations. The task in this thesis was to study how to produce the material data required in the Fatemi and Socie critical plane model. A parameter handling the normal stress influence on the crack initiation life was especially studied.

The study was performed under the assumption that results from torsion tests and tension tests could be utilized to represent the new data. First, structural analyses were performed in the commercial FE-code ANSYS for specimens exposed to tension and torsion, respectively. The stress states achieved were then used in crack initiation analyses in the in-house fatigue computer program fatsoc8.f90. The results show an increasing normal stress influence with increasing crack initiation life. However, the method utilized is very sensitive to changes and requires exact input data for the results to be reliable. A suitable continuation would be to perform a crack initiation analysis of a component, subjected to multiaxial and non-proportional loading, applying the new data and compare to experimental results. In addition, it would be interesting to investigate the possibility to use two different tension tests for calculation of the parameter.

iv

Contents

1. INTRODUCTION......................................................................................................... 1

1.1 BACKGROUND ............................................................................................................ 1 1.2 TASK DESCRIPTION..................................................................................................... 1 1.3 METHOD .................................................................................................................... 2

2. THEORY ....................................................................................................................... 3

2.1 CRYSTALLINITY ......................................................................................................... 3 2.2 FATIGUE..................................................................................................................... 3

2.2.1 Stress-life approach ........................................................................................... 3 2.2.2 Strain-life approach ........................................................................................... 4 2.2.3 Rain Flow Counting........................................................................................... 5 2.2.4 Damage sum....................................................................................................... 7

2.3 CRITICAL PLANE MODELS........................................................................................... 7 2.3.1 The Fatemi and Socie critical plane model ....................................................... 7

2.3.1.1 What is k? ................................................................................................... 9

3. TORSION TESTS....................................................................................................... 10

4. STRUCTURAL ANALYSIS...................................................................................... 13

4.1 ANSYS MECHANICAL ............................................................................................. 13 4.2. TORSION ANALYSIS ................................................................................................. 13 4.3 TENSION-COMPRESSION ANALYSIS .......................................................................... 15 4.4 RESULTS .................................................................................................................. 16

4.4.1 Torsion results ................................................................................................. 16 4.4.2 Tension results ................................................................................................. 17

5. CRACK INITIATION ANALYSIS .......................................................................... 19

5.1 FATSOC8.F90............................................................................................................ 19 5.2 METHOD .................................................................................................................. 22 5.3 RESULTS .................................................................................................................. 22 5.4 USING THE RESULTS: TRANSFORMING FROM RΕ=-1 TO RΕ=0 ................................... 30

6. DISCUSSION AND CONCLUSIONS ...................................................................... 33

7. FUTURE WORK........................................................................................................ 35

REFERENCES................................................................................................................ 36

NOMENCLATURE........................................................................................................ 37

Appendix 1 Torsion specimen

Appendix 2 Tension-compression specimen

Appendix 3 *.seq files

1

1. Introduction

1.1 Background

Volvo Aero Corporation (VAC) develops and manufactures components for jet engines and rocket engines. Fatigue life predictions are essential and are partially performed by means of crack initiation analyses. A crack initiation analysis predicts the number of load sequences that can be applied to a component before a small crack has been initiated. With a small crack is often meant a crack, for which it is possible to apply linear fracture mechanics and compute crack propagation. However, the definition is somewhat diffuse. Today, so called one-parameter models are most often used to model this initiation. In one-parameter models, one parameter is calculated from and represents the stress- and strain state. This parameter can for instance be maximum principal strain or maximum shear stress. Early fatigue problems were investigated by testing. Since the problems arose in structures with mainly uniaxial loading, test procedures were developed on the basis of uniaxiality. When complicated loading situations thereafter had to be investigated, it was natural to use already existing uniaxial test data. Thus, in one-parameter models a general multiaxial stress state is translated into a uniaxial stress measure. One-parameter models work well enough for conditions close to uniaxiality and for situations where the different stresses and strains applied, respectively, differ only by a factor, so called proportional loading. However, recent research shows that these models are not sufficient for strongly multiaxial and non-proportional loading conditions. During the last two decades so called critical plane models have been developed. The aim of these models is to better describe multiaxiality and non-proportional loading. One significant difference between these new models and older one-parameter models is that two, instead of one, parameters are representing the stress state.

1.2 Task description

Volvo Aero Corporation is currently working with the implementation of the Fatemi and Socie critical plane model into in-house fatigue computer programs. As a result, new material data will be needed in addition to already existing uniaxial data. The aim of this thesis is to investigate how this new material data should be produced. An assumption is that torsion test results in addition to existing uniaxial tension-compression test results would be sufficient for extracting the new data. For this reason, specimens have been produced and exposed to torsion testing. This thesis will therefore focus on the evaluation of the recently performed torsion tests, but also on the evaluation of older tension-compression test results. The expected result is a work description of how to produce material data for the Fatemi and Socie model. The description should explain how to complement existing uniaxial data as well as how to proceed with no data available at start.

2

The material studied is a nickel-base alloy. For this material Volvo Aero has got quite a lot of uniaxial data.

1.3 Method

The work was divided into the following parts:

• Literature survey concerning fatigue concepts and critical plane models, especially the Fatemi and Socie model.

• Perform ANSYS structural analyses of specimens exposed to tension-compression and torsion, respectively, in order to obtain the stress –and strain states.

• Use results from the ANSYS analyses and perform fatigue analyses with fatsoc8.f90, which is a prototype program for crack initiation analysis based upon the Fatemi and Socie critical plane model. This will yield values for the two parameters of the model

• Evaluate the above analyses and develop a work description of how to produce new material data to the Fatemi and Socie model.

• If there is still time, start preparing for a crack initiation analysis and test of a component subjected to multiaxial and non-proportional loading, with this new material data applied to it. The analysis will thereafter be compared to the experimental results.

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2. Theory

2.1 Crystallinity

Metallic materials are crystalline, which means that the atoms are arranged in repeating three-dimensional patterns. The most common crystal types are body-centred cubic (BCC), face-centred cubic (FCC) and hexagonal closed packed. (HCP). The crystals may have defects or imperfections. A dislocation is a linear defect around which the atomic pattern is changed. Plastic deformation of materials arises when dislocations are set in motion, causing atomic planes in the crystal to slip. The dislocations most easily move in planes and directions with dense atomic packing. These planes and directions are called slip planes and slip directions. The slip plane together with its corresponding slip direction constitutes a slip system. The number and orientation of the slip systems depend on the crystal type.

2.2 Fatigue

Fatigue is a form of failure that occurs in structures exposed to repeated loading. It is responsible for approximately 90% of all failures in metal structures [1]. Fatigue analyses of components and structures are therefore of utmost importance. The fatigue process is divided into three different steps: crack initiation, crack propagation and final failure. In the first step, a small crack is initiated. Most often this occurs in areas with high stress concentrations. The crack then propagates with every stress cycle until a critical size is reached and failure occurs, often very rapidly. This thesis focuses on the crack initiation part of the fatigue process. Very often turbojet- and rocket engine components are designed to meet crack initiation criteria only.

2.2.1 Stress-life approach

An important term within fatigue is the fatigue life, which states the ability of a component to withstand exposure to cyclic loads. The fatigue life is commonly expressed as number of cycles to failure. For a load sequence, where the loading amplitude is varying, the fatigue life of a component should be expressed as number of load sequences to failure. To illustrate fatigue properties, applied stress amplitude is often plotted versus the logarithm of the numbers of cycles to failure, resulting in so called S-N diagrams, see figure 2.1. Also frequent is to plot the logarithm of the strain range versus the logarithm of the number of cycles to failure. Fatigue life is also used to distinguish between low-cycle fatigue (LCF) and high-cycle fatigue (HCF). LCF usually includes cycles to failure below 10 000 to 100 000 cycles. Characteristic for LCF is that the peak stresses are above the tensile yield strength, resulting in plastic straining [2]. This is one important failure mechanism in jet engines. In HCF, the stresses and strains are confined to the elastic region resulting in longer lives.

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Figure 2.1 An example of an S-N curve.

2.2.2 Strain-life approach

In many components the material response is strain dependent. Therefore, a strain-life approach exists, in which the fatigue life is determined by the strain range. In 1910, O.H. Basquin observed that the former mentioned S-N data could be expressed as:

b

f N )2(2

' ⋅⋅=∆

σσ

(2.1)

and thereby linearised with respect to the logarithms of the left hand side and the right

hand side of equation (2.1). '

fσ is the fatigue strength coefficient and b is the fatigue

strength exponent. Equation (2.1) corresponds to the elastic range of the material,

thus22

eE εσ ∆⋅=

∆, according to Hooke´s law.

In 1954, L.F Coffin and S.S Manson found that plastic strain life data could be expressed as:

c

f

pN )2(

2

' ⋅⋅=∆

εε

, (2.2)

where '

fε is the fatigue ductility coefficient and c is the fatigue ductility exponent. This

equation also becomes a straight line in a log-log domain. Since the total strain is the sum of the elastic and plastic strain, an expression relating strain to fatigue life may now be written as:

c

f

bfNN

E)2()2(

2

'

'

⋅⋅+⋅⋅=∆

εσε

(2.3)

Str

ess a

mp

litu

de

103 10

4 10

5 10

6 10

7 10

8 10

9 10

10

Cycles to failure, N

5

This expression is known as the Basquin-Coffin-Manson equation or just the Coffin-Manson equation.

2.2.3 Rain Flow Counting

In FE-analysis, performed for instance in ANSYS, the continuous load sequence must be discretised. The load sequence is hence divided into a number of parts, termed load steps. In the following description, it is assumed that the loading situation has been discretised. When applying a load sequence on a component, not every load step contributes to the fatigue. Only the turning points, that is, the peaks and troughs in a stress plot, are likely to cause damage. Rain Flow Counting is a method to extract these damaging cycles. The applied load sequence results in an array of varying stress measures, figure 2.2 a). As a first step, this array is arranged so that it starts and ends with the largest positive stress value. This is accomplished by dividing the array between the largest positive stress value and the previous element. The latter part of the array is then moved to the front, without changing the order of the elements (b).As a result, this new array has one more element than the previous one. Thereafter, the array is searched for consecutive element doublets, which are removed. Elements not causing turning points and hence not contribute to the damage are also removed (c). The array is then searched and a cycle is counted each time the stress range for successive peaks and troughs is larger than the former range (d). It is the former range that constitutes the range of the counted cycle. This cycle is recorded, with range and mean, and its peak and trough is removed from the array. The procedure is repeated until only one cycle from the original load sequence remains and this cycle is removed as well (e). The cycles extracted, (f), are then further processed before their fatigue lives finally can be calculated. If the FE-analysis was linearly elastic, an elastic-plastic correction normally has to be done. A method for elastic-plastic correction is the Neuber rule, which states that:

Knn

elastic

e

elastic

e =⋅=⋅ σεσε , (2.4)

where elastic

eε and elastic

eσ are the strains and stresses obtained through Rain Flow Counting

and nε and nσ are the elastic-plastic corrected strains and stresses. A

certain elastic

eε together with its corresponding value of elastic

eσ results in a Neuber

hyperbola ε

σK

= . The corresponding values of nε and nσ are found as the intersection

point between this hyperbola and the stress-strain curve obtained from tension tests.

6

Since the fatigue behaviour is affected by the mean stress in a cycle, it is also necessary to perform a mean stress correction. An example of a mean stress correction method is Morrow’s approach. In this case, the Basquin-Coffin-Manson curve is modified according to:

c

f

bmfNN

E)2()2(

)(

2

'

'

⋅⋅+⋅⋅−

=∆

εσσε

,

where mσ is the mean stress. (2.5)

Finally the fatigue lives for the extracted cycles can be determined using equation (2.5) and the elastic-plastic corrected stresses and strains. Rain Flow Counting is further described in [3].

Figure 2.2 Rain Flow Counting.

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2.2.4 Damage sum

When the fatigue lives for the cycles have been found, the Palmgren-Miner rule is used to sum the damages from the cycles. For each cycle j, the corresponding damage is the inverse of the number of cycles to failure:

jj

ND

1= . (2.6)

The Palmgren-Miner rule gives the damage sum as:

∑∑ == jDNj

D1

. (2.7)

The inverse of D, 1/D, then gives the total fatigue life for the whole load sequence.

2.3 Critical plane models

One tool for crack initiation analyses within fatigue is critical plane models. This concept has been developed during the last thirty years and aims to improve fatigue life predictions in multiaxial and non-proportional loading situations. The first model was presented in the 1970´s by K. Miller and M. Brown, see [4]. Today, several different critical plane models exist. One significant difference between these new models and older one-parameter models is that two, instead of one, parameters are representing the stress state. Critical plane models not only predict fatigue life, but also determine the dominant failure plane. A plane containing the point of analysis is selected and for each load step in the load sequence, the two parameters (different for different critical plane models) are extracted to represent the stress state. Thereafter, Rain Flow Counting is performed with respect to the two parameters, hence these parameters corresponds to σeff in figure 2.2. The results are compared to method-specific material data in order to obtain the fatigue lives for the damaging extracted cycles. Finally, the Palmgren-Miner rule gives the damage sum, D, for the whole load sequence. The procedure is then repeated for a number of different planes and that plane resulting in the largest damage sum is thus the critical plane.

2.3.1 The Fatemi and Socie critical plane model

The Fatemi and Socie critical plane model was presented in 1988. The model is as follows [4]:

γγ γτ

σ

σγ c

f

bf

yield

nNN

Gk )2()2(1

2

´

´

max,⋅+⋅=

+

∆. (2.8)

8

In this model, the shear strain range ∆γ and the maximum normal stress σn,max in a plane are the two parameters to be considered. Shear loading is essential for crack initiation since fatigue cracks initiate on planes of high shear [5]. According to [4] the normal stress is also important. A positive normal stress increases the distance between the atomic planes in the structure, which facilitates the shear loading to cause damage. The opposite occurs for a negative normal stress. Also included in the left hand side of equation (2.8) are the yield strength σyield and a parameter k, see chapter 2.3.1.1. The left hand side of equation (2.8) is denoted “effective shear strain amplitude”:

+

∆

yield

nk

σ

σγ max,1

2= effective shear strain amplitude.

The right hand side contains the number of cycles to crack initiation, N, and some material dependent constants:

´

fτ is the shear fatigue strength coefficient

γb is the shear fatigue strength exponent ´

fγ is the shear fatigue ductility coefficient

γc is the shear fatigue ductility exponent

G is the shear modulus The Fatemi and Socie model is based on the Basquin-Coffin-Manson equation and the two equations also resemble each other. A good approximation of the constants in the Fatemi and Socie equation is [4]:

3

'

´ f

f

στ ≈

3'´ ⋅≈ ff εγ

bb ≈γ cc ≈γ

When the two parameters, ∆γ and σ n,max, have been extracted for each load step j in the load sequence for a specific plane, Rain Flow Counting is performed with respect to the left hand side of equation (2.8).

Hence,

jyield

nk

+

∆

σ

σγ max,1

2corresponds to σeff in figure 2.2. The fatigue life for each

extracted cycle is then calculated using the right hand side of equation (2.8) or its graphical or tabulated representation. The Palmgren-Miner rule gives the damage sum for the load sequence and the inverse, 1/D, is the fatigue life for the point of analysis in that plane.

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2.3.1.1 What is k?

The left hand side of equation (2.8) contains a parameter k, known as the normal stress sensitivity. This parameter describes the influence of normal stress on fatigue life. Today, no information about k is available at Volvo Aero Corporation. One aim of this thesis is to investigate how to determine k so that the Fatemi and Socie model can be utilized. It has been shown [4], that k is material- and life dependent and the latter relationship will be studied thoroughly in the thesis. The basis for the study can be related to the foundation of the Fatemi and Socie model: If two constant amplitude tests with different states of stress result in the same fatigue

life, they must have equal “effective shear strain amplitude”.

The tension-compression and torsion tests will therefore be investigated for the same crack initiation life, N. When the left hand side of the Fatemi and Socie equation is equal for the two kinds of tests, a correct value of k has been achieved. In the crack initiation analysis, the procedure will therefore be to vary k until this is achieved. If no value of k fulfils the criteria, torsion testing might not be the correct approach. In [4], the k-N relationship has been investigated for an HSLA steel with E = 210 GPa. The result is extracted from [4] and shown in figure 2.3. Although k is material dependent the diagram might be useful when studying the relationship for the nickel-base alloy.

Figure 2.3 k-N relationship for an HSLA steel according to [4].

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3. Torsion tests As mentioned earlier, specimens exposed to torsion will be the focus of this thesis. The material studied is a nickel-base alloy with σy =364MPa and E = 205GPa. The torsion tests were performed at the Royal Institute of Technology in Stockholm. Figure 3.1 is a picture of the test arrangement. In order to measure the twisting, two fixtures were placed 45mm apart at each specimen, see figure 3.2. The edges of the fixtures were situated 25mm outside the axis of symmetry of the specimen. As the specimen was twisted, the relative displacement of the edges was measured with a clip gage. This displacement was denoted ∆CG and the corresponding angle of twist was

calculated using the formula 25

CG∆=α , with ∆CG in mm.

Three different load levels were to be applied to the test specimens and there were three to four specimens for each load level, see table 3.1. The load for each level was varied between plus and minus the same magnitude (R-value R=-1). The load levels could be represented by shear stresses, τ. The torque corresponding to each shear stress was calculated as:

WM ⋅= τ , (3.1)

where W= )(16

44 dDD

−⋅π

and D and d are the outer and inner diameter of the

specimen, respectively. The angle of twist, α, caused by the torque was thereafter determined using the clip gage. However, for the highest load level α was too large for the clip gage to detect. Instead, the angle of twist of the piston was measured, denoted νrot. Results are shown in table 3.1.

Figure 3.1 The test arrangement.

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Figure 3.2 Specimen with fixtures attached.

Specimen τ [MPa] Number of cycles to

failure

νrot [°] ∆CG [mm] α [°]

1 240 194 800 ± 2.85 ± 0.68 ± 1.56

2 240 187 700 ± 2.68 ± 0.61 ± 1.40

3 240 446 000 ± 2.73 ± 0.62 ± 1.42

4 276 21 330 ± 4.35 ± 1.21 ± 2.77

5 276 14 700 ± 4.09 ± 1.05 ± 2.41

6 276 23 000 ± 4.69 ± 1.17 ± 2.68

7 330 10 290 ± 5.62

8 330 3 240 ± 10.1

9 330 4 010 ± 10.0

10 330 3 215 ± 10.0 Table 3.1 Results from the torsion tests.

Only one tension-compression and one torsion analysis, respectively, will be performed at each stress level. Therefore, mean values of the variables from table 3.1 will be employed. However, since the results for specimen 7 clearly differ from the other they will not be included. To obtain the value of α for the highest stress level the Royal Institute of Technology have estimated ∆CG in the following way: νrot is a function of the twisting of the cylindrical part of the specimen and the twisting of the specimen and test machine outside the cylindrical part. Hence,

MyCGxrot ⋅+∆⋅=ν , (3.1)

where M is the torque. By means of a least square fit, the values for x and y are 3.4 and 0.0109, respectively. The three values of ∆CG for specimens 8,9 and 10 are then 2.755mm, 2.735mm and 2.73mm. Table 3.2 displays the mean values for the different stress levels.

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τ [MPa] Number of cycles

to failure

νrot [°] ∆CG [mm] α [°]

240 276166.7 ± 2.75 ± 0.637 1.46

276 19676.7 ± 4.38 ± 1.143 2.62

330 3488.3 ± 10.0 ± 2.74 6.28 Table 3.2 Mean values of the torsion tests.

13

4. Structural analysis Structural analyses are performed in order to obtain the stress –and strain states for the tension-compression and torsion specimens, respectively. The shear strain range ∆γ and maximum normal stress σn,max in the Fatemi and Socie critical plane model cannot be measured directly through specimen tests, but has to be calculated from the stresses and strains obtained in an FE-analysis.

4.1 ANSYS Mechanical

The structural analyses were performed with ANSYS Mechanical 10.0. This is a Finite Element Analysis (FEA) software containing pre-processor, solver and post-processor. In FEA, the structure to be analysed is divided into a mesh of finite elements. For each element, the software implements and solves equations that govern its behaviour. Combined, these results give a description of how the structure operates as a whole. The procedure in ANSYS can be divided into the following parts: build the geometry, define material properties, generate mesh, apply loads, obtain solution and display results. The following chapters describe how the structural analyses were performed for the tension-compression and torsion test specimens, respectively.

4.2. Torsion analysis

The specimens to be modeled are hollow with (at their thinner part) an inner radius varying between 5.74 and 5.75mm and an outer radius varying between 6.67 and 6.675 mm. To simplify, mean values are set to 5.74mm and 6.67mm. The specimens have a length of approximately 184mm. Appendix 1 contains a picture of the specimen. PLANE25 is an axisymmetric 2-D element type often used in structural analyses. However, in this case it was not possible to use this element type since it cannot handle elastic-plastic behaviour. Instead, the element type selected for the analysis is SOLID95. This is a 20-node 3-D solid element with three degrees of freedom per node: translation in the x, y, and z directions. SOLID95 can handle plasticity, creep, stress stiffening, large deflections and large strains. Due to the element type selected it is necessary to perform 3-D simulations. Because of axial symmetry, a 5° slice of the specimen is sufficient. In addition, only 100mm of the specimen is modeled. This length includes the thinner part of the specimen plus the transition to the thicker part. In the tests, mainly the thinner part is affected. To define material properties, a material data file containing variables such as modulus of elasticity and Poisson’s ratio is added to the script. In order to achieve a nonlinear model, a table for kinematic hardening is imported.

14

During the simulations a mesh with 2592 elements is used. The mesh is finer at the thinner part of the model. Figure 4.1 displays half of the mesh.

Figure 4.1 Half the mesh used

in the torsion analyses.

Anti-symmetric boundary conditions are applied at the model boundaries at 0° and 5°. The displacements of the nodes located at one of the thicker ends of the specimen are locked. A global cylindrical coordinate system is used to specify the boundary conditions for nodes attached to the other end. These conditions are displacements in the tangential direction, which will cause the specimen to twist. Four load steps are used in the analyses. First, the specimen is twisted at angle α, then back to the start position, next twisted at angle - α and finally back to the start position. Trial and error is used to find the displacement that will lead to the correct α. The nodes

are given a certain displacement expressed as β⋅= rd , where r is the distance to the axis

of symmetry for a specific node and β is an arbitrary angle expressed in radians, equal for all nodes. The analysis is then performed and in the post-processing two nodes located 45mm apart at the outer radius of the model are selected. These nodes correspond to the fixtures in the torsion tests. Their relative displacement in the first load step is determined and α is calculated. The procedure is repeated with varying β until the correct value of α is found. Table 4.1 shows how to choose β for the different values of α.

15

α.[°] β [°]

1.46 2.22

2.62 3.87

6.28 8.94 Table 4.1 β applied to achieve the different

angles of twist α.

However, before the calculations are started, analysis type and load step options are defined. In this case, static analyses are performed with automatic time stepping within each load step.

4.3 Tension-Compression analysis

A typical specimen for tension-compression tests is shown in Appendix 2. The specimen has a length of 97mm and its radius varies between 3.175mm and 9.525mm. For the same reasons as for the torsion specimen, only the thinnest cylindrical part needs to be analysed. Due to symmetry, a 2-D cut of one fourth of the cylinder is sufficient. The same material data file and kinematic hardening table utilized for torsion are utilized also here. The element type used in the analysis is Plane82. This is an 8-node element with two degrees of freedom: translation in the x and y directions. The element can handle plasticity, creep, swelling, stress stiffening, large deflections, and large strains. During the simulations a mesh with 226 elements is used. The elements are given an axisymmetric behaviour. Figure 4.2 displays the mesh.

Figure 4.2 The mesh used in the

tension-compression analyses.

The specimen is exposed to tension and compression in four load steps: first tension, then back to the start position, after that compression and finally once more back to the start position. To achieve the same number of cycles to crack initiation as for the torsion test,

16

these tests have to generate special strain ranges. The ε∆ -N relationship for the nickel-base alloy can be found in Volvo Aeros material data base MVISION (curve number M1008LF6A). Table 4.2 shows the strain ranges used in the tests.

N ε∆ [%]

3488.3 1.75

19676.7 0.93

276166.7 0.375 Table 4.2 Strain ranges for the tension tests.

The nodes located at the upper edge of the specimen are locked. For the lower edge, the nodes are coupled and given a displacement that will generate the correct strain range for each test. Half the strain range is achieved through tension and the other half through

compression. Hence, 1max

min −=ε

ε. The

max

min

ε

ε ratio is denoted the R-value, Rε.

As in the torsion analyses, static analyses are performed with automatic time stepping within each load step.

4.4 Results

4.4.1 Torsion results

Figure 4.3 shows the von Mises stress for the first load step for the specimen with average crack initiation fatigue life N=19676.7. As stated in chapter 4.2, the stress field is almost homogenous in the cylindrical part. Similar stress fields (but with different magnitudes) are obtained for the other two simulations.

The stresses and strains needed for the crack initiation analyses are evaluated for one node, no. 4244. The node is situated at the outside of the specimen and 22.86mm below the center – an area representative for that length area of the test specimens where all cracks initiated. Its approximate position is shown in figure 4.3. The normal stresses, shear stresses, normal strains and shear strains for all load steps are gathered in a file, *.seq, that fatsoc8.f90 will make use of, see Appendix 3.

17

Figure 4.3 von Mises stress for average crack initiation fatigue

life N=19676.7. Torsional loading, first load step.

4.4.2 Tension results

Figure 4.4 displays the von Mises stress for the first load step for the specimen with average crack initiation fatigue life N=3488.3. The stress field is practically homogenous in the thinnest cylindrical part. As seen before, the result is similar for the other analyses. Node no. 89 is chosen for the stresses- and strain evaluations and is also shown in figure 4.4. It is located at the outside of the specimen, 1.5mm above the center. Appendix 3 shows the *.seq files.

18

Figure 4.4 von Mises stress for average crack initiation fatigue

life N=3488.3. Tensional loading, first load step.

19

5. Crack initiation analysis

5.1 fatsoc8.f90

In order to obtain values for σn,max and ∆γ, crack initiation analyses are performed using fatsoc8.f90. This is a prototype program where the Fatemi and Socie critical plane model is implemented. When using a critical plane model, the calculations tend to be extensive. A number of different planes are evaluated and for each plane, several different directions are studied to find the largest effective shear strain range. VAC has developed a specific technique to make the calculations more efficient. For a full description, see [6]. The first step in the analysis is to transform the stresses and strains (obtained in the structural analysis) from the global coordinate system x, y, z to the coordinate system x’, y’,z’ defining the plane investigated, see figure 5.1. In this case, the global coordinate system corresponds to θ=0° and Φ=90°. The transformation is defined as follows [4]:

⋅

+++

+++

+++=

yz

xz

xy

z

y

x

zy

zx

yx

z

y

x

aaaaaaaaaaaaaaaaaa

aaaaaaaaaaaaaaaaaa

aaaaaaaaaaaaaaaaaa

aaaaaaaaa

aaaaaaaaa

aaaaaaaaa

τ

τ

τ

σ

σ

σ

τ

τ

τ

σ

σ

σ

)()()(

)()()(

)()()(

222

222

222

322333223321312331223221332332223121

331232133311311331123211331332123111

221323122311211321122211231322122111

323333313231

2

33

2

32

2

31

222323212221

2

23

2

22

2

21

121313111211

2

13

2

12

2

11

''

''

''

'

'

'

⋅

+++

+++

+++=

2/

2/

2/

)()()(

)()()(

)()()(

222

222

222

2/

2/

2/

322333223321312331223221332332223121

331232133311311331123211331332123111

221323122311211321122211231322122111

323333313231

2

33

2

32

2

31

222323212221

2

23

2

22

2

21

121313111211

2

13

2

12

2

11

''

''

''

'

'

'

yz

xz

xy

z

y

x

zy

zx

yx

z

y

x

aaaaaaaaaaaaaaaaaa

aaaaaaaaaaaaaaaaaa

aaaaaaaaaaaaaaaaaa

aaaaaaaaa

aaaaaaaaa

aaaaaaaaa

γ

γ

γ

ε

ε

ε

γ

γ

γ

ε

ε

ε

where

φ

φθ

φθ

cos

sinsin

sincos

13

12

11

=

⋅=

⋅=

a

a

a

0

cos

sin

23

22

21

=

=

−=

a

a

a

θ

θ

φ

φθ

φθ

sin

cossin

coscos

33

32

31

=

⋅−=

⋅−=

a

a

a

To find the damage sum and hence the fatigue life for a component for a specific plane, it is necessary to calculate the maximum normal stress and shear strain for each load step in

20

a load sequence. This will result in normal stress σn max i and shear strains γx’y’i and γx’z’i . The plot of the shear strains in the plane is called the gammapath, see figure 5.1. The next step is to find the direction in the plane, causing the largest damage and transform the shear strains into that direction. As mentioned in chapter 2.1 dislocations move easiest in the slip directions of the crystal. Therefore, the largest damage will be obtained in one of these directions. BCC-crystals have their slip directions at 0° and 90°, FCC- and HCP crystals at 0°, 60° and 120°. Assuming all crystals have their slip directions at 0° and 90°, the maximum angle error will be 30°. Transforming the shear strains into those directions causes a maximum error of 13.4% for the shear strain values, cos(30°) = 0.866. This approximation is assumed to be good enough. Thus, the calculations are reduced to evaluate damage along the two, perpendicular slip directions. To find the two directions, an inertial analogy is utilized. The analogy states that the two directions of interest coincide with the two principal axes of inertia for those steps in the gamma path representing turning points. A turning point is a step for which the next and the previous step both have smaller or both have greater absolute value of γx

,y, or γx

,z,. A

“weight” value of

⋅+

yield

ixk

σ

σ ,'1 is assigned to each turning point. The value is thought of

as a “point mass”, that is

⋅+=

yield

ix

i kmσ

σ ,'1 . (5.1)

The total mass for all turning points are ∑= imM . (5.2)

The distance of the point masses to the coordinate axes y’ and z’ are

γx’z’i and γx’y’i, respectively. According to [7] the angle between the principal axes of inertia and the coordinate axes η and ζ can be calculated as:

−

⋅⋅=

ηζ

ηζψII

I2arctan5.0 , (5.3)

where ζI and ηI are the moments of inertia around the “centre of gravity axis” η and ζ,

respectively, and ηζI is the product of inertia around the same axes. η and ζ are y’-and z’-

axes translated to the centre of gravity for the total mass M. First, the centre of gravity in y’ and z’ coordinates are calculated:

M

Sy z '' = ,

M

Sz

y '' = , (5.4), (5.5)

where Sy’ and Sz’ are the moments around y’ and z’.

21

∑=

⋅=n

i

iiyxz mS1

''' γ , ∑=

⋅=n

i

iizxy mS1

''' γ (5.6), (5.7)

Thereafter, the moments of inertia and the product of inertia around the y’ and z’ axes are calculated:

∑=

⋅=n

i

iizxy mI1

2

''' γ , ∑=

⋅=n

i

iiyxz mI1

2

''' γ , ∑=

⋅⋅=n

i

iiyxizxzy mI1

'''''' γγ (5.8), (5.9), (5.10)

The moments of inertia and the product of inertia around the η -and ζ-axes are then found using Steiner’s theorem:

MzII y ⋅−= 2'

' )(η ,)( 2'

' MyII z ⋅−=ζ MyzII zy ⋅⋅−= ''

''ηζ (5.11), (5.12), (5.13)

Putting (5.11), (5.12), (5.13) in equation (5.3) gives the sought angles ψ1 and ψ2. The gamma path is then translated into the new coordinate system y’’ and z’’, defined by ψ1 and ψ2, resulting in new values γx’’y’’ and γx’’z’’ for the shear strains. Rain Flow Counting

is then performed on the “weighted” values

⋅+

yield

ix

iyx kσ

σγ ,''

'''' 1 and

⋅+

yield

ix

izx kσ

σγ ,''

'''' 1 .

The counting results in a number of extracted cycles (normally different numbers for y’’ and z’’) with maximum normal stress σmax x’’j and shear strain ranges ∆γx’’y’’j and ∆γx’’z’’j . For each cycle j, the “effective shear strain amplitude” is calculated:

⋅+

∆

yield

jxjyxk

σ

σγ ,''max''''1

2and

⋅+

∆

yield

jxjzxk

σ

σγ ,''max''''1

2.

Together with the right hand side of equation (2.8) these give the fatigue lives, Ny’’j and Nz’’j, for each extracted cycle, with respect to y’’ and z’’. The damage sum for each direction is calculated using the Palmgren-Miner rule:

∑=jy

yN

D''

''

1 and ∑=

jz

zN

D''

''

1.

The damage sum for the plane is found as the sum of these:

'''' zyplane DDD += .

22

Figure 5.1 A critical plane with coordinate system x’y’z’,

defined by θ and Φ. The gamma path is expressed in

shear strain values γx,y,i and γx

,z,i.

5.2 Method

To perform a crack initiation analysis, three files are needed, *.seq from the ANSYS analysis, *.info which states the names of the input files and the solution files to be

written and finally *.inp. In *.inp, yieldσ is determined and the θ and Φ values, for which a

plane is investigated, are controlled. In the analyses, θ and Φ are investigated for every 2.5° - at 0°, 2.5°, 5° etc. A value of k has to be chosen as well. For a specific N, the same value of k should be chosen for the tension-compression and torsion analyses, respectively. fatsoc8.f90 will then calculate σn,max, ∆γ , the “effective shear strain amplitude” and the damage sum. The “effective shear strain amplitude” is calculated with respect to y’’ and z’’ and thus two different values will be available. However, only the largest of them (corresponding to the most damaging direction) will be studied. When the“effective shear strain amplitudes” for the tension-compression and torsion analyses are equal, the correct value of k has been found. In order to achieve a value of N, the constants from the right hand side of equation (2.8) should also be stated in the *.inp file. However, since N already is known in this case and only the “effective shear strain amplitudes” are of interest, these constants are not defined here.

5.3 Results

Figure 5.2 and 5.3 are examples of how the total damage varies for the different planes studied in the tension-compression and torsion analyses, respectively. In these cases, average crack initiation fatigue life is N=3488.3 and k is set to 0.40. The largest damage is found for the plane represented by θ=127.5° and Φ=102.5° in the tension-compression

23

analysis and by θ=90° and Φ=172.5° in the torsion analysis. Hence, these are the critical plane directions. In both cases, the total damage is approximately 0.0004, which is expected since the fatigue life is equal. The critical plane direction found in the torsion analysis is close to the actual crack initiation direction seen in figure 3.2, where the crack has started approximately at θ=90°and Φ=180° and propagated mainly in this direction. Also for the tension-compression test there is a good agreement between predicted angle for crack initiation and actual crack initiation angle. Both figures 5.2 and 5.3 indicate that the damage varies significantly between different planes. Hence, finding the critical plane is essential in order to predict the correct fatigue life. In figure 5.2, the damage varies with θ although the load is pure axial. This is explained by the orientation of the specimen (see figure 4.2). In these analyses, the y-axis corresponds to the axial direction and θ is consequently not the tangential coordinate (see figure 5.1). Also seen i figure 5.2 is that certain planes give zero damage. These are planes where the shear strain ranges are small.

Figure 5.2 Total damage for the different planes studied in the

tension-compression analyses. Average crack initiation fatigue

life N=3488.3 and k=0.40.

24

Figure 5.3 Total damage for the different planes studied in the

torsion analyses. Average crack initiation fatigue life N=3488.3

and k=0.40.

Figure 5.4, 5.5 and 5.6 show the “effective shear strain amplitude”,

+

∆

yield

nk

σ

σγ max,1

2, as a

function of k for the three different values of N analysed. k has been varied in a range from 0-2 to study where the “effective shear strain amplitudes” are equal and thus obtain the correct value of k. k is approximately 0.44 for N=3488.3 and 0 for N=19676.7, whereas no value of k has been found for N=276166.

25

0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60

0.0160

0.0165

0.0170

0.0175

0.0180

0.0185

0.0190

k

eff

ective s

hear

str

ain

am

plit

ude

Figure 5.4 “Effective shear strain amplitude”-k relationship for

N=3488.3. The dashed line represents the torsion results and

the dotted line the tension-compression results.

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

0.0070

0.0075

0.0080

0.0085

0.0090

k

eff

ective s

hear

str

ain

am

plit

ude


N=19676.7 The dashed line represents the torsion results and


26

0.5 1 1.5 20.75 1.25 1.75

0.0035

0.0040

0.0045

0.0050

0.0055

0.0030

k

eff

ective s

hear

str

ain

am

plit

ude


N=276166.7 The dashed line represents the torsion results and


In figure 5.7, the “effective shear strain amplitude” is shown as a function of N for k=0.40. Although k is constant in figure 5.7 but fatigue life dependent in reality, the results describe the behaviour well. A smaller change in k will not significantly affect the results. As mentioned in chapter 2.3.1 the Fatemi and Socie model resembles the Basquin-Coffin-Manson equation (2.3). Since the Basquin-Coffin-Manson equation can be linearised in a log-log plot it is likely that this is possible for the Fatemi and Socie equation as well. However, figure 5.7 implies that this can only be achieved for the tension-compression results. Therefore the logarithmic torsion results are adjusted using a least squares fit. The procedure is presented in figure 5.8. Similar adjustments are made for values of k varying between 0.2 and 1.5, a range in which the correct values of k are likely to be found.

27

103

104

105

106

10-3

10-2

10-1

N

eff

ective s

hear

str

ain

am

plit

ude

Figure 5.7 “Effective shear strain amplitude” as a function of N

for k=0.40. * represent the torsion results and o represents the

tension-compression results.

3.5 4.0 4.5 5.0 5.5

-2.4

-2.3

-2.2

-2.1

-1.9

-2.0

-1.8

-1.7

log(N)

log(e

ffective s

hear

str

ain

am

plit

ude)

Figure 5.8 Least square fit of the torsion results for k=0.40.

* represents the logarithmic torsion results and the line

shows the adjustment.

With the new values of the torsion “effective shear strain amplitudes”, a value of k for each N now can be found. k is approximately 0.20 for N=3488.3, 0.445 for N=19676.7 and 1.17 for N=276166.7, see figure 5.9, 5.10 and 5.11.

28

0.20 0.40 0.60 0.80 1.0

0.014

0.016

0.018

0.020

0.022

k

eff

ective s

hear

str

ain

am

plit

ude


N=3488.3 after the adjustments. The dashed line represents the

torsion results and the dotted line the tension-compression results.

0 0.2 0.4 0.6 0.8 1.0

0.0075

0.0080

0.0085

0.0090

0.0095

0.0100

0.0105

0.0110

0.0115

k

eff

ective s

hear

str

ain

am

plit

ude




29

0.6 0.8 1.0 1.2 1.4

0.0032

0.0034

0.0036

0.0038

0.0040

0.0042

0.0044

0.0046

k

eff

ective s

hear

str

ain

am

plit

ude




The k-N relationship is plotted in figure 5.12 together with the relationship from figure 2.3. Clearly, the range of k is much larger in this study than for the HSLA steel between N=3488.3 and N=276166.7; 0.20-1.17 compared to 0.3-0.5. The value of k for N=276166.7 differs the most. The result in this study indicates a k-value of 1.17 and hence a normal stress influence of 117%.

Figure 5.12 The k-N relationship. * represents the results from

this study and the line represents the results for an HSLA steel

according to [4].

30

However, the results for N=3488.3 are somewhat uncertain. In this case, the torsion tests were controlled by the motion of the piston instead of the clip gauge and the angle of twist was obtain by means of a least squares fit, equation (3.1)

MyCGxrot ⋅+∆⋅=ν .

For the other two values of N, this fit generated angles of twist that differed up to 4.5% from the result obtained in the tests. To investigate how such a difference affects the results, the angle of twist α for N=3488.3 was adjusted 4% from 6.28° to 6.53°. A new torsion structural analysis was performed using this angle of twist and once again the “effective shear strain”-k relationship was investigated for the three values of N. As before, the logarithm of the torsion results did not generate a linear relationship and a least squares fit was done, similar to the one in figure 5.8. Finally, the k-N relationship in figure 5.13 was obtained. The previous results with 6.28° as the largest angle of twist are also included. As shown, the value of k for N=276166.7 has decreased considerably, resulting in a smaller range.

Figure 5.13 The k-N relationship. * represents the results

with 6.28° as largest angle of twist and ∆ represents

the new results with 6.53° as largest angle of twist.

5.4 Using the results: Transforming from Rε=-1 to Rε=0

In order to investigate the reliability of the results obtained, new tension analyses are performed. This time Rε=0, which indicates that the analyses are solely tensile in nature. The strain ranges applied in the structural analyses are 0.00375, 0.0060 and 0.0093, respectively. In the analyses, only two load steps are used: first tension to achieve the correct strain range and then back to the start position. However, in the first load step some plastic deformation is induced in the specimens modeled. Therefore a little compression has to be added in the second load step to achieve zero strain.

31

Figure 5.14 displays the “effective shear strain amplitude” as a function of N as found in figure 5.9-5.11. This relationship is implemented in fatsoc8.f90 to yield values of N once the “effective shear strain amplitudes” have been calculated for Rε=0. The implementation is a first attempt to study the reliability of the k-N relationship obtained. Figure 5.15 is a plot of the k-N relationship for the nickel-base alloy from figure 5.12. The results have been interpolated between the three measured values.

103

104

105

106

0.001

0.01

0.1

N

eff

ective s

hear

str

ain

am

plit

ude

Figure 5.14 The “effective shear strain amplitude”-N

relationship implemented in fatsoc8.f90.

Figure 5.15 The k-N relationship obtained in the

previous study.

32

For each strain range in the Rε=0 analysis, a value of k is chosen and the corresponding value of N from figure 5.15 is noted. In fatsoc8.f90 the “effective shear strain amplitude” is calculated and a value of N is obtained. The correct k is found when the two values of N are equal and the procedure is repeated until this is achieved. The results, presented as strain range versus N, are shown in figure 5.16. The corresponding results for Rε=-1 obtained from Volvo Aeros material data base MVISION are also included. As shown, Rε=0 gives a linear relationship in the loglog plot. For a specific strain range, the number of cycles to crack initiation is less for Rε=0 compared to Rε=-1.

103

104

105

106

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

1.3

N

tota

l str

ain

range (

%)

Figure 5.16 Total strain range versus N. o represents

Rε=0 and the line Rε=-1.

33

6. Discussion and conclusions The aim to find a relationship between the normal stress sensitivity parameter, k, and the number of cycles to crack initiation, N, has been reached, see figure 5.12. The method to analyse tension-compression and torsion stress states, respectively, seems to be a correct approach. However, for the results to be reliable certain changes and improvements of the work procedure described in this thesis have to be done. The results in figure 5.16, which were produced in order to study the reliability of the results achieved, seem realistic. For a specific strain range, the number of cycles to crack initiation is lower for Rε=0 than for Rε=-1. This is due to a higher mean stress for Rε=0 during the load sequence. It can also be noted that this difference in N is decreasing with increasing strain range. This is reasonable, since high strain ranges result in plastic deformation which often lowers the mean stress. Since there are only three values of N studied it is difficult to make detailed conclusions about the k-N relationship. More results are needed in order to explain the behaviour of k between these values. For higher values of N it is important to investigate if k is still increasing or if there possibly is a limit. Compared to figure 2.3, the range of k is much higher in this study. Since the materials are not the same, it is difficult to tell whether this is correct or not. However, one noted relationship is that k is increasing with increasing values of N, which agrees with the behaviour in figure 2.3. This indicates that the normal stress influence increases with increasing values of N. One important aspect during this research is how sensitive the method is to changes in the input data. Especially the torsion results were affected considerably, as can be noted in figure 5.13. When changing the angle of twist by 0.25°, the range of k decreased from 0.97 to 0.59. This observation indicates that the torsion input data used needs to be exact in order to obtain useful results. For the tension-compression structural analyses however, a change in the strain range for each N barely affected the results. The sensitivity requires the torsion tests to be performed more accurately. Primarily, it is important to measure the angle of twist with extreme precision. For the largest stress level the movement of the piston was observed and the angle of twist calculated through a least squares fit, resulting only in approximate values. To obtain more exact values, some kind of technical device, as used for the lower stress levels, is preferable. Uncertainties in the angle of twist might be the reason why the torsion effective shear strain amplitudes had to be adjusted, which was carried out in chapter 5.3. In addition, it is important to make clear that it is the number of cycles to crack initiation that are of interest, not the number of cycles to failure. In these tests, crack initiation was detected by studying when the load starts to drop. The numbers of cycles to initiation and failure, respectively, did not differ a great deal, but for other tests the difference might be significant. Another way to measure crack initiation could for instance be to pressurize the specimens. When the pressure starts to decrease, a crack has been initiated.

34

During this work, a new idea turned up. Instead of using tension and torsion data to find k, it should be possible to use tension data for Rε=0 and Rε=-1, respectively. The work procedure would be the same as in this thesis; to obtain equal “effective shear strain amplitudes” for equal values of N. This method is very appealing since torsion tests are approximately twice as expensive as tension-compression tests. The difference is mainly due to a higher cost to manufacture the torsion specimens. In addition, the torsion tests must be performed by other companies since no equipment is available at Volvo Aero. Therefore, it is much more time-consuming to gather torsion data than tension data. If the new idea works, two methods would be available to obtain k. However, it is possible that the methods would produce different values for k. If this turns out to be the case, then one should choose method depending on the stress state to be analysed. For example, if shear stresses are of interest the method involving torsion data is useful. Otherwise, the method involving pure tension data is to recommend considering cost and time.

35

7. Future work It is important to further study the reliability of the results achieved. A suitable continuation would be to perform a crack initiation analysis of a component applying the new data and compare to experimental results. In addition, it would be interesting to study the possibility to use tension tests with Rε=-1 and Rε=0, respectively, to produce the material data.

Furthermore, one has to decide if fatsoc8.f90 should make use of the constants ´

fτ , bγ, ´

fγ

and cγ in the Fatemi and Socie critical plane model or use an “effective shear strain amplitude-N” relationship in order to calculate the crack initiation lives. The latter method was used in this thesis. If the former method is chosen, one must investigate how to produce the constants, since these are not yet identified.

36

References [1] Callister, William D. Jr (2003) Materials Science and Engineering An Introduction.

John Wiley & Sons, Inc. ISBN: 0-471-22471-5 [2] Conway, Joseph B., Sjodahl, Lars H. (1991) Analysis and Representation of Fatigue

Data. ASM International. ISBN: 0-87170-427-7

[3] Khosrovaneh, A.K, Dowling, N.E (1990) Fatigue loading history reconstruction based on the rainflow technique. International Journal of Fatigue, 12 (2), 99-106

[4] Socie, Darrell F., Marquis, Gary B. (2000) Multiaxial Fatigue. SAE, Inc. ISBN:0-7680-0453-5 [5] Bannantine, Julie A., Comer, Jess J., Handrock, James L. (1990) Fundamentals of

metal fatigue analysis. Prentice-Hall Inc. ISBN:0-13-340191-X [6] Samuelsson, Leif, Rosheden, Tommy (2006) Critical Plane: Effective Methods for

Localisation and Load Cycle Evaluation. (Volvo Aero Corporation) 9:th International Fatigue Congress 14-19 May 2006, Atlanta USA, Ref no: FT 25 [7] Sundström, B. et al. (1998) Handbok och formelsamling i Hållfasthetslära, KTH.

37

Nomenclature

E modulus of elasticity G shear modulus

Rε max

min

ε

ε

yσ yield strength

N number of cycles to crack initiation

σ∆ stress range

ε∆ strain range

γ∆ shear strain range

max,nσ maximum normal stress

b fatigue strength exponent c fatigue ductility exponent

'

fε fatigue ductility coefficient

'

fσ fatigue strength coefficient

´

fτ shear fatigue strength coefficient

´

fγ shear fatigue ductility coefficient

bγ shear fatigue strength exponent cγ shear fatigue ductility exponent load sequence consists of all loads in the correct order they are applied to a component load step the different loads of the load sequence

38

non-proportional load for most of the load steps in a load sequence the stresses and strains in different steps are not differing only by a factor proportional load for all load steps in a load sequence the stresses and strains in different steps are differing only by a factor multiaxial stress state two or three principal stresses have equal order of magnitude

Appendix 1

Torsion specimen

Appendix 2

Tension-compression specimen

¨

Appendix 3

*.seq files Note that the files have been divided at the time column.

Vrid_N3488.3.seq

NUMBER OF NODES = 1

NUMBER OF L.STEPS = 5

NODE NO = 4244

SXX SYY SZZ SXY SYZ SXZ TEMP TIME

0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 293.000 0.0000

-0.22548 0.11776 -0.17491 -0.14078E-03 0.30922E+09 -41.422 293.000 1.0000

0.91271E-02 0.22077E-01 -0.69950E-01 0.15342E-01 -0.22315E+09 7.9071 293.000 2.0000

0.22039 -0.22463E-01 0.15999 0.29783E-02 -0.30922E+09 45.051 293.000 3.0000

-0.14521E-01 -0.15249E-01 0.59876E-01 -0.13153E-01 -0.22315E+09 -5.7789 293.000 4.0000

TIME EXX EYY EZZ EXY EYZ EXZ

0.0000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000

1.0000 -0.27146E-11 0.24182E-11 -0.21632E-12 -0.16812E-12 0.16186E-01 -0.19118E-08

2.0000 -0.10254E-11 0.18171E-11 -0.86193E-12 0.50810E-12 0.45009E-03 -0.89854E-09

3.0000 0.24598E-11 -0.29781E-11 0.11678E-11 0.71203E-12 -0.16185E-01 0.16935E-08

4.0000 0.81931E-12 -0.22992E-11 0.15346E-11 0.21084E-14 -0.44990E-03 0.75708E-09

Vrid_N19676.7.seq

NUMBER OF NODES = 1


NODE NO = 4244


0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 293.000 0.0000

-0.10529 -0.62818E-01 -0.12261E-01 -0.62022E-02 0.25453E+09 -19.616 293.000 1.0000

0.17050E-01 -0.64889E-02 0.53151E-02 0.31618E-02 -0.14188E+09 -1.5268 293.000 2.0000

0.10498 0.57806E-01 0.13379E-01 0.74647E-02 -0.25453E+09 20.611 293.000 3.0000

-0.17575E-01 0.33872E-02 -0.49154E-02 -0.22990E-02 0.14188E+09 2.3240 293.000 4.0000


0.0000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000

1.0000 -0.69434E-12 -0.11892E-12 0.48600E-12 -0.16776E-12 0.67687E-02 -0.48903E-09

2.0000 -0.12799E-12 -0.10392E-12 0.26071E-12 -0.20838E-13 0.27447E-03 -0.23355E-09

3.0000 0.65640E-12 0.43370E-13 -0.38014E-12 0.21303E-12 -0.67687E-02 0.43843E-09

4.0000 0.94131E-13 0.38260E-13 -0.16705E-12 0.62224E-13 -0.27448E-03 0.19246E-09

Vrid_N276166.7.seq

NUMBER OF NODES = 1


NODE NO = 4244


0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 293.000 0.0000

-0.68981E-01 -0.31974E-01 -0.10223E-01 -0.41867E-02 0.21184E+09 -10.358 293.000 1.0000

0.66212E-02 -0.82947E-02 0.42480E-02 -0.96906E-03 -0.62748E+08 -1.4315 293.000 2.0000

0.69143E-01 0.32568E-01 0.10263E-01 0.44890E-02 -0.21184E+09 10.500 293.000 3.0000

-0.64892E-02 0.88171E-02 -0.42474E-02 0.12362E-02 0.62748E+08 1.5620 293.000 4.0000


0.0000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000

1.0000 -0.34822E-12 -0.21250E-13 0.16775E-12 -0.75555E-13 0.37562E-02 -0.18038E-09

2.0000 -0.29899E-13 -0.48458E-13 0.83028E-13 -0.33361E-13 0.10778E-03 -0.63590E-10

3.0000 0.34347E-12 0.16918E-13 -0.15722E-12 0.78400E-13 -0.37562E-02 0.17357E-09

4.0000 0.25435E-13 0.44466E-13 -0.73384E-13 0.36076E-13 -0.10779E-03 0.57307E-10

Drag_N3488.3

NUMBER OF NODES = 1


NODE NO = 89


0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 293.000 0.0000

-16466. 0.52610E+09 0.24141E+06 -71643. 0.0000 0.0000 293.000 1.0000

3957.4 -0.33936E+09 -41103. 8698.6 0.0000 0.0000 293.000 2.0000

24686. -0.52620E+09 -0.44088E+06 75875. 0.0000 0.0000 293.000 3.0000

-675.62 0.33931E+09 -38384. -6605.4 0.0000 0.0000 293.000 4.0000


0.0000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000

1.0000 -0.39063E-02 0.87550E-02 -0.38937E-02 -0.21792E-05 0.0000 0.0000

2.0000 -0.77847E-03 0.93207E-03 -0.76940E-03 -0.80164E-06 0.0000 0.0000

3.0000 0.39059E-02 -0.87561E-02 0.38947E-02 0.24438E-05 0.0000 0.0000

4.0000 0.77841E-03 -0.93308E-03 0.77024E-03 0.10265E-05 0.0000 0.0000

Drag_N19676.7

NUMBER OF NODES = 1


NODE NO = 89


0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 293.000 0.0000

-13423. 0.45625E+09 0.24983E+06 -18372. 0.0000 0.0000 293.000 1.0000

-11499. -0.22352E+09 0.31093E+06 19325. 0.0000 0.0000 293.000 2.0000

17544. -0.45634E+09 -0.35945E+06 12166. 0.0000 0.0000 293.000 3.0000

14512. 0.22345E+09 -0.38869E+06 -23735. 0.0000 0.0000 293.000 4.0000


0.0000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000

1.0000 -0.19194E-02 0.46639E-02 -0.19163E-02 -0.63567E-06 0.0000 0.0000

2.0000 -0.49883E-03 0.58838E-03 -0.49456E-03 -0.21252E-08 0.0000 0.0000

3.0000 0.19202E-02 -0.46651E-02 0.19163E-02 0.60596E-06 0.0000 0.0000

4.0000 0.49965E-03 -0.58949E-03 0.49459E-03 -0.33608E-07 0.0000 0.0000

Drag_N276166.7

NUMBER OF NODES = 1


NODE NO = 89


0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 293.000 0.0000

1440.7 0.32636E+09 -80354. -16829. 0.0000 0.0000 293.000 1.0000

-291.64 -0.36356E+08 -9471.7 -3440.9 0.0000 0.0000 293.000 2.0000

-1256.5 -0.32637E+09 75105. 16323. 0.0000 0.0000 293.000 3.0000

470.49 0.36349E+08 4502.4 2967.2 0.0000 0.0000 293.000 4.0000


0.0000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000

1.0000 -0.64278E-03 0.18782E-02 -0.64343E-03 -0.26189E-06 0.0000 0.0000

2.0000 -0.80469E-04 0.95128E-04 -0.80641E-04 -0.89133E-07 0.0000 0.0000

3.0000 0.64282E-03 -0.18783E-02 0.64342E-03 0.25244E-06 0.0000 0.0000

4.0000 0.80501E-04 -0.95176E-04 0.80636E-04 0.80153E-07 0.0000 0.0000

material data derivation for the fatemi and socie …1021821/fulltext01.pdfimprove fatigue life...

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