Continuous Time Fatigue Modelling for Non-proportional Loading

Linköpings universitet SE–581 83 Linköping

Linköping University | Department of Management and Engineering

Master’s thesis, 30 ECTS | Aeronautical Engineering

2019 | LIU-IEI-TEK-A—19/03610—SE

Continuous Time Fatigue

Modelling for Non-proportional

Loading

Sajjan Gundmi Satish

Supervisor : Stefan Lindström Examiner : Peter Schmidt

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Abstract

Fatigue analysis is a critical stage in the design of any structural component. Typically fatigue is analysed during post-processing, but as the size of the analysed component in-creases, the amount of data stored for the analysis increases simultaneously. This increases the computational and memory requirements of the system, intensifying the work load on the engineer. A continuum mechanics approach namely ’Continuous time fatigue model’, for fatigue analysis is available in a prior study which reduces the computational require-ments by simultaneously computing fatigue along with the stress. This model implerequire-ments a moving endurance surface in the stress space along with the damage evolution equation to compute high-cycle fatigue.

In this thesis the continuous time fatigue model is compared with conventional model (ie. Cycle counting) to study its feasibility. The thesis also aims to investigate the continuous time fatigue model and an evolved version of the model is developed for non-proportional load cases to identify its limitations and benefits.

Acknowledgments

My sincere gratitude goes to my supervisor Stefan Lindström for giving me this opportunity to conduct a study in the field of my passion. His reliable presence and guidance throughout the thesis was vital to produce a well rounded work. I would like to thank my examiner Peter Schmidt for his valuable feedback and support.

I would also like to give heartfelt thanks to my family and friends for motivating and sup-porting me in this journey.

Linköping, March 2020 Sajjan

Contents

Abstract iii

Acknowledgments iv

Contents v

List of Figures vi

List of Tables vii

1 Introduction 1 1.1 Motivation . . . 1 1.2 Aim . . . 1 1.3 Research questions . . . 1 1.4 Delimitation . . . 2 1.5 Matlab . . . 2 2 Theory 3 2.1 Stress-strain analysis . . . 3 2.2 Fatigue Analysis . . . 4 2.3 Model Concept . . . 6

2.4 Evolved Model Concept . . . 9

3 Implementation 12 3.1 Model Development . . . 12

3.2 Validation . . . 13

3.3 Calibration of Parameters . . . 15

4 Case Study and Results 18 4.1 Comparison with Cycle Counting Methods . . . 18

4.2 Validity Range for Non-proportional Loading . . . 20

4.3 Validity Range of the Evolved Model for Non-proportional loading . . . 22

5 Discussion 26 5.1 Results . . . 26

5.2 Impact . . . 27

6 Conclusion 28

List of Figures

2.1 Arbitrary body under external load . . . 3

2.2 Stress components defined in an infinitesimal volume element . . . 5

2.3 Uniaxial and biaxial loading of a cylinder . . . 5

2.4 S-N curve for a material with a fatigue limit of 490 MPa . . . 6

2.5 Endurance surface when α=0 in deviatoric plane . . . 8

2.6 Endurance surface when α ‰ 0 in deviatoric plane . . . . 8

2.7 Evolution of D and α for loading and unloading in the deviatoric plane . . . . 9

2.8 Evolved model in deviatoric plane . . . 10

2.9 Ramp function, R(x) . . . 11

2.10 Updated Ramp function, S(x) . . . 11

3.1 Algorithms . . . 12

3.2 Flow Chart of the Odefn Function . . . 13

3.3 Development of periodic movement of the Endurance surface . . . 14

3.4 Endurance limit predication in Haigh diagram . . . 15

3.5 Wöhler Curve between initial and calibrated parameters at σm=0 MPa . . . 16

3.6 Wöhler Curve between initial and calibrated parameters at σm=100 MPa . . . 16

3.7 Wöhler Curve between initial and calibrated parameters at σm=200 MPa . . . 16

3.8 Wöhler Curve for the calibrated parameters at various σm. . . 17

4.1 Undulating Load Cycle . . . 19

4.2 Zoomed in for clearer view . . . 19

4.3 Endurance Surface development for the undulating load (Zoomed in for clearer view) . . . 20

4.4 Stress Path for Eq.(4.3) . . . 21

4.5 Damage development for Eq.(4.3) . . . 21

4.6 Damage development for Tension-shear Load for the CTF model . . . 22

4.7 Stress path for combination of tension-shear load . . . 22

4.8 Damage development of the evolved model . . . 23

4.9 Damage development for various values of a at φ=π/2 . . . 24

4.10 Wöhler Curve for various values of a . . . 24

4.11 Wöhler curve for the calibrated parameters σm=0 . . . 25

4.12 Wöhler curve for the calibrated parameters σm=100 MPa (left) and σm =200 MPa (right) . . . 25

List of Tables

4.1 Cycle counting results for a small sample . . . 19 4.2 Calibrated parameters . . . 23

1

Introduction

1.1

Motivation

The cyclic loading of a material causes it to fail at much lower stresses than when monoton-ically loaded, a phenomenon known as fatigue [4]. Fatigue analysis is a critical stage in the design of any mechanical structure. Generally, fatigue load histories are complex and call for cycle-counting methods to obtain an equivalent cycle to ascertain damage [17]. These conventional methods of analyzing fatigue require cyclic loads, however in the real world, the loading is often non-cyclic and possibly non-proportional which complicates the use of cycle counting methods. Fatigue analysis is primarily separated into two regimes, high-cycle fatigue, where the stress is low and primarily elastic and failure require more than 104 cy-cles, and low-cycle fatigue where the stress is high and there is significant plasticity. The continuous time fatigue (CTF) model [13] provides us with the concept of moving endurance surfaces in the stress space, which is analyzed together with damage evolution equation for high cycle fatigue analysis. The CTF model considers crack initiation in its formulation. This concept performs well under multiaxial proportional load conditions, but the influence of non-proportional loading on the structure is yet to be studied. Currently, there exists new unpublished work which evolves the continuous time model further for the analysis of ma-terial which lack a distinct endurance limit (e.g. Aluminium).

1.2

Aim

Scientific studies for continuous time fatigue model so far have been conducted for propor-tional loading and to a very limited extent on non-proporpropor-tional loading. However, in order to implement the model in the FEM software, the model must accurately predict fatigue for all conceivable stress cases. Hence, the purpose of this thesis is to analyze the influence of biaxial, non-proportional loading on this model to assess its validity range. The study will also explore and validate an evolved model that addresses the issues identified in the original model.

1.3

Research questions

1.4. Delimitation

1. How does the continuous time fatigue model compare with the conventional model (Cycle count-ing)?

2. What is the validity range of the continuous time fatigue model for biaxial non-proportional loading?

The validity range refers to the set of stress fluctuations that results in stable realistic outputs that the system must produce when compared to the experimental results. 3. What is the validity range of the evolved model for biaxial non-proportional loading?

The validity range of the evolved model is studied to see if it expands upon the CTF model.

1.4

Delimitation

The delimitations of this analysis method are as follows: • The material in the study is considered to be isotropic.

• Material and geometry specific properties such as surface roughness and stress gradi-ents are neglected.

• While crack initiation is considered, crack propagation is excluded from the study. • Plasticity due to accumulated damage is neglected. The model is linear-elastic and is

limited to high-cycle fatigue.

• Statistical aspects of fatigue are neglected to avoid excessive data handling and to con-duct the study within the limited time.

• Temperature effects on the material under study is ignored • Environmental effects such as corrosion are neglected.

1.5

Matlab

The analysis will be conducted in Matlab and the various inbuilt functions such as ode45 to solve the initial value problem, fminsearch for solving the optimization problem, will be used. It would be beyond the scope of this thesis to manually create these mathematical functions. These functions will be considered to work optimally and will not be checked for accuracy. The handbook from Mathworks will be used as a guide to understand these functions [11].

2

Theory

2.1

Stress-strain analysis

In this section the general theory of stress and strain will be explored. When a component has an external force exerted on it, it gives rise to internal forces to maintain equilibrium. These internal forces are manifested as stress and strain, where stress is the force per unit area exerted by the particles and strain quantifies the deformation. Elasticity is defined as the ability of a body under external loads to return to its original shape when the load is removed.

2.2. Fatigue Analysis

When an arbitrary body fig. (2.1) is loaded the intensity at infinitesimally small region within it is given by

lim

AÑ0

F

A =t (2.1)

Where, F is the force, A is the area and t is the traction vector. The traction vector relates the normal n= (n1, n2, n3)in the Cartesian coordinates and the stress tensor σ by, t=σ.n. Stress

state gives us the current stresses in a body due to loads at any time. This stress is described using the Cauchy stress tensor as

σ=   σ11 σ12 σ13 σ21 σ22 σ23 σ31 σ32 σ33   (2.2)

where i,j=1,2,3 of σij is related to the coordinate planes as shown in an infinitesimally small

volume element, fig. (2.2). The first number denotes the surface normal and the second denotes the direction of the stress. Equilibrium of moments of a volume element implies that Eq. (2.2) is symmetric such that

σ12=σ21, σ13 =σ31, σ23=σ32 (2.3)

From Eq. (2.2) and (2.3) we can see that there are 9 stress components acting on the body, out of which 3 normal and 3 shear forces (each represented twice) are present. To simplify this the Voigt notation is used henceforth:

σσσ=         σ11 σ22 σ33 σ12 σ13 σ23         (2.4)

2.2

Fatigue Analysis

The fatigue damage in a component can be classified into the following stages: (1) nucleation and formation of micro-cracks; (2) coalescence of micro-cracks causing creation and stable growth of macro-cracks; (3) unstable macro-crack propagation and fracture [19]. The creation of these micro-cracks are influenced by various factors like surface roughness, environmental conditions, size of the component, defects in the micro-structure such as voids and stress gradients due to manufacturing [4, 10]. Fatigue design involves estimation of fatigue life in terms of cyclic stress range or strain range when designing against fatigue crack initiation or by analysing the time it takes to propagate from a initial crack size to some critical dimension. A few comparisons of existing fatigue models can be found in [20, 15]. High-cycle fatigue (HCF) typically occur at lower stresses for which the time for crack initiation is higher than 104_{cycles. Stress based approaches are commonly used for HCF and are primarily divided}

into three main groups, stress invariant criteria [18, 2, 3], critical plane criteria [12, 5] and average stress criterion [9, 14]. The CTF model is based on the critical plane criteria.

Cyclic Loading

The continuous and repeated application of loads on a material or component causes fatigue. Cyclic loading causes deterioration of material at much lower stresses than for constant load-ing. The cyclic loads can be uniaxial or multiaxial. Uniaxial loading refers to loading of the

2.2. Fatigue Analysis

Figure 2.2: Stress components defined in an infinitesimal volume element

component in one axis, biaxial loading refers to loading of the component in two axes and multiaxial loading also includes loading in 3 or more axes. The loading for instance can be tension, compression or torsion which are common in experimental investigations of fatigue.

Figure 2.3: Uniaxial and biaxial loading of a cylinder

Proportional and Non-proportional Loading

When a component is biaxially or multiaxially loaded, the cyclic loads can have a lag or a phase difference between them. When the loads act simultaneously with no phase difference the component is said to be under proportional loading. If there is a phase difference between the loads then it is said to be under non-proportional loading.

2.3. Model Concept

S-N curves

Wöhler in 1860 proposed experiments to predict the fatigue life of materials. ASTM E466-E468 provide protocols to obtain the fatigue life in terms of nominal stress amplitudes. In these experiments the stress amplitude σa for a fully reversed uniaxial loading is plotted

against the number of cycles to fatigue failure, N, and this is called the S-N curve. Under con-stant loading conditions the materials exhibit a plateau in the S-N curve, below which it the material may be cycled indefinitely without failure, fig. (2.4). This limiting stress amplitude is know as the endurance limit or fatigue limit, σe.

100 105 1010 1015 Number of cycles N 450 500 550 600 650 700 750 Stress in [MPa]

Figure 2.4: S-N curve for a material with a fatigue limit of 490 MPa

Cumulative Damage

In reality the components often have complex loading with varying stress amplitudes, mean stresses and loading frequencies. This requires cycle counting methods like rain-flow count-ing to obtain blocks of different stress amplitude and calculatcount-ing damage, D, for each cycle [19]. This damage is summed over the entire load history and it varies between 0 and 1, where 0 is undamaged and 1 is critical failure. The Palmgren-Miner rule gives a method to calculate accumulated damage, as

D=ÿ i

ni

Ni

(2.5) where, ni is the number of cycles for a specific block of constant stress amplitude and Ni is

the number of cycles to failure for that cycle type.

2.3

Model Concept

Ottosen et al [13] provide a model for high cycle fatigue for arbitrary multiaxial loading. The concept of a moving endurance surface with a damage evolution equation is utilized. The endurance function is given by β(σ, α)and the endurance surface is defined by β = 0. Movement of the endurance surface is modelled with a deviatoric stress which defines the center of the endurance surface. Damage evolution is registered whenever the stress state is outside the endurance surface, and the time rate of endurance function is positive. The model treats uniaxial and multiaxial stress states in one framework for arbitrary load histories thus avoiding cycle-counting techniques.

2.3. Model Concept

Tensor Definitions

The model utilizes various tensors and tensor operators. These operator definitions are ex-plained here:

Deviatoric Stress

The deviatoric of stress tensor (s) is obtained by subtracting the hydrostatic stress tensor from the Cauchy stress tensor (σ) Eq.(2.2), given by,

s=dev(σ) and dev(σ) =σ ´1

3tr(σ)δ (2.6)

where the δ is the unit tensor and tr(σ) is the first invariant of stress tensor, or trace, given by the sum of the diagonal elements of a symmetric stress tensor.

Forbenius Norm

The Forbenius norm denoted by ||x|| is defined as

||x||=ax : x (2.7)

The : operator is defined as the Forbenius inner product for tensors x and y is given by x : y=tr(x yT).

The following theroem from [7] is also required. For a deviatoric tensor x, tr(x) = 0, and y is time differentiable, then

x : d dtdev(y) =x : d dt  y ´1 3tr(y)δ  =x : ˙y ´1 3tr(˙y)(x : ˙ δ) =x : ˙y (2.8) where x :δ˙= tr(x) = 0.

Governing Equations

From [8], for a given load cycle, the stress history is given by σ(t), where t is the time. The back stress α is a deviatoric tensor and β is the endurance function.

β(σ, α) = 1 σe

[σe f f(σ, α) +Atr(σ)´ σe] (2.9)

where σe is the endurance stress or fatigue limit, A is a material constant and the effective

stress, σe f f, is given by

σe f f(σ, α) =

c 3

2||s ´ α|| (2.10)

The damage accumulation is dependent on this deviatoric stress s. σe f f defines the effective

stress modified by the back-stress α. The endurance surface is defined by β = 0. The stress state inside the endurance surface is given by β ă 0 and outside when β ą 0, respectively. The fig. (2.5) and fig. (2.6) shows the endurance surface for α=0 and α ‰ 0 respectively. The evolution of the back-stress α allows the endurance surface to move in the stress space.

2.3. Model Concept

Figure 2.5: Endurance surface when α=0 in deviatoric plane

Figure 2.6: Endurance surface when α ‰ 0 in deviatoric plane

0 (undamaged) to D = 1 (critical failure). The damage evolution equations are given by the ordinary differential equations (ODEs).

˙

α= (s ´ α)CH(β)H(˙β)˙β (2.11)

˙

D=g(β)H(β)H(˙β)˙β (2.12)

where C>0 and g(β) =Kexp(Lβ), where K and L are dimensionless positive material param-eters. The ˙α and ˙D are the time derivatives and H denotes the Heavside step function defined as,

H(x) =

#

1 if x ą 0

0 otherwise (2.13)

As (2.11) and (2.12) are dependent on ˙β, direct time differentiation of (2.9) is performed. How-ever ˙β obtains a ˙α term, to remove this dependency we substitute (2.11) to ˙β and we find

σe˙β= 3 2 (s ´ α) σ_{e f f}(σ, α) : ˙s ´ σe f f (σ, α)CH(β)H(˙β)˙β+Atr(σ)˙ (2.14) Simplifying we get [σe+σe f f(σ, α)CH(β)H(˙β)]˙β= 3 2 (s ´ α) σe f f(σ, α) : ˙s+Atr(σ)˙ (2.15) and by defining a rate function,

ν(σ, ˙σ, α) = 1 σe+Cσe f f(σ, α) " 3 2 (s ´ α) σe f f(σ, α) : ˙s+Atr(σ)˙ # (2.16)

From (2.15) we an see that, H(ν) = H( ˙β), and H( ˙β) ˙β = H(ν)ν, thus the damage evolution equa-tion can be simplified as follows,

˙

α= (s ´ α)CH(β)H(ν)ν (2.17)

˙

2.4. Evolved Model Concept

The fig. (2.7) shows the movement of the endurance surface. Wherein the endurance surface will track the stress point and the direction of the movement is given by(s ´ α).

There are five material parameters (σe, A, C, K, L) which have to be calibrated. The equations

Figure 2.7: Evolution of D and α for loading and unloading in the deviatoric plane (2.17),(2.9),(2.10) and (2.16) can be used to solve the back stress evolution explicitly using Dormand-Prince method via ode45 in Matlab [11].

2.4

Evolved Model Concept

The evolved model is derived from [7], and it proposes a geometric framework for the equa-tions in the Ottosen model. It utilizes two characteristic unit tensors for the given stress states (σ, ˙σ), they are in the gradient direction u and the loading direction v as seen in the fig. (2.8) These gradients are defined as

u= Bβ Bσ › › › › Bβ Bσ › › › › ´1 = b 1 3 2+3A2 c 3 2 s ´ α ||s ´ α||+Aδ ! (2.19) v=σ ||˙ σ||˙ ´1 (2.20)

Also using Eq.(2.10) and theorem (2.8) the rate function Eq.(2.16), can be written as

ν(σ, ˙σ, α) = 1 σe+ b 3 2C||s ´ α|| c 3 2 s ´ α ||s ´ α||+Aδ ! (2.21)

Substituting Eq.(2.19) and Eq.(2.20) into Eq.(2.21), the rate function ν can be written as,

ν=F(σ, α)(u : v)|| ˙σ||

σe (2.22)

where function F is given by,

F(σ, α) = b 3 2+3A2 1+ b 3 2 C σe||s ´ α|| ą0 (2.23)

2.4. Evolved Model Concept

Consequently

R(ν) =F(σ, α)R(u : v)|| ˙σ||

σe (2.24)

Therefore the evolved model for back-stress and damage development can be obtained by

Figure 2.8: Evolved model in deviatoric plane inserting Eq.(2.24) into Eq.(2.17) and Eq.(2.18), we get

˙ α=F(σ, α)(s ´ α)CH(β)R(u : v)|| ˙σ|| σe , α(0) =0, (2.25) ˙ D=F(σ, α)g(β)H(β)R(u : v)|| ˙σ|| σe , D (0) =0. (2.26) The ramp function, R governs the functional dependence on the cosine between the gradient and loading direction, u : v. The evolved model utilizes the geometric interpretation of the model in second order tensor space. The fig. (2.8) shows the how the gradient and loading direction are represented in the deviatoric plane.

R(x) =

#

x if x ą 0

2.4. Evolved Model Concept

Figure 2.9: Ramp function, R(x)

Figure 2.10: Updated Ramp function, S(x)

The dashed line in fig. (2.8) shows the arbitrary stress path, but if this stress path becomes rotary in nature then the gradient and loading directions will become perpendicular to each other, u : v « 0. This implies that there is no damage evolution for such a stress state. To solve this we implement an updated ramp function S(u : v;a), such that the function steps up before becoming perpendicular. The a here is the step for the ramp function. The action of the Ramp function R and S can be seen in the fig. (2.9) and fig. (2.10).

S(x) =

#

x+a

1+a if x ą a

3

Implementation

3.1

Model Development

The governing equations will be implemented in Matlab to compute the back-stress ˙α and damage evolved ˙D for a load cycle in the steady state which can then be utilized to compute the fatigue life.

(a) Main Function

(b) Life Function

3.2. Validation

Main Function

The main function is primarily used to input the material parameters, load cycles for which fatigue life is to be calculated. It also performs post processing to read and visualize the results obtained.

Life function

This function takes the material parameters from the main function, initializes α and D and calls the ODE solver (ode45 in Matlab). This solver calls the function "odefn" to compute ˙α and ˙D. The damage developed is used to compute the fatigue life of the material by consid-ering the increment in damage per cycle to be in constant steady state, we use the final and penultimate peaks to obtain damage incurred and its reciprocal gives the fatigue life.

ODE function

The ODE functions uses Matlab’s inbuilt code, ode45, to compute the ˙α and ˙D over an inter-val of time. Here, mean stress and stress amplitude is used to create the stress cycle at the particular instance. It also contains inline functions to compute the trace, deviatoric of tensor and also the Forbenius product and Forbenius norm as defined in section 2.3. These functions are used to solve equations (2.10), (2.9), (2.16) and then (2.17) and (2.18) for the original model and these values are updated for each instance.

Figure 3.2: Flow Chart of the Odefn Function

3.2

Validation

Before moving further it is important to make sure that the model is implemented correctly. For this we utilize the integrated solution for SAE 4340 subjected to tension/compression

3.3. Calibration of Parameters

[13] and recreate the periodic movement of the endurance surface and the damage evolution under uniaxial loading. It is assumed that α and σ for some t is given by

α=   α 0 0 0 ´1₂α 0 0 0 ´1₂α 

for some unknown α(t) (3.1)

and σ=   σ 0 0 0 0 0 0 0 0  for time-varying σ(t) (3.2)

For this investigation we use NiCrMo steel SAE 4340 whose endurance limit is, σe= 490 MPa.

We consider it to be loaded with a mean stress, σm=0 MPa and stress amplitude, σa= 600 MPa.

The material parameters from [13] are A = 0.225, C = 1.25, K = 2.65E-5 and L = 14.4. The periodic movement can be analyzed by first considering σ and α which reduces (2.10) to

σe f f =κ  σ ´3 2α  and κ= # 1 if σ ´3₂α ą0 ´1 if σ ´3₂α ă0 (3.3)

Eq.(2.9) then takes the form,

β= 1 σe  κ  σ ´3 2α  +Aσ ´ σe  (3.4) If we solve (3.4) for β=0 we obtain

σκ =

κ3₂α+σe

A+κ (3.5)

which represents the endurance surface in this one-dimensional case. Solving Eq. (2.17) give us α=α11and this is used with the stress Eq. (3.5) to record the load path where the damage

is developed, fig. (3.3).

3.3. Calibration of Parameters

0 100 200 300 400 500 600 700

Mean Stress in [MPa] 100 200 300 400 500 600 700 800

Stress Amplitude [MPa]

Yield Limit

Endurance limit data points Endurance Limit

Figure 3.4: Endurance limit predication in Haigh diagram

3.3

Calibration of Parameters

There are five material parameters which need to be calibrated. The parameters σeand A can

be calibrated by using uniaxial load cases for different mean stresses and endurance limits. The endurance limit at mean stresses of 0, 100, 200, 400 MPa is used to plot the endurance limit curve, whose y-intercept gives the endurance limit (σe) and the (negative) slope gives

A. The yield stress (σy) of the material is used as the limiting factor such that the data is only

considered if σm+σa ď σy, this is represented as the yield limit in fig. (3.4). The fit of this

curve, fig. (3.4), is obtained using only the first three data point as mean stress of 400 MPa lies outside the yield limit.

Calibration of the remaining parameters is done by calculating the error in the fatigue life predicted by the model against experimental data. Due to the complexity of calibration, L is fixed at 14.4 and while C and K are fit in a least-square sense. At least 3 data points are required to solve the minimization problem. The experimental data for each mean stress contains 5 data points, but the final data point is omitted as it is the endurance limit which was already used for fitting σeand A and also as the prediction error in N diverges to infinity

at the endurance limit. The data points for σm = 400 MPa are excluded as they lie outside

the yield limit as seen in fig. (3.4) and cannot be considered in HCF analysis. We compute the fatigue life for each of these data points and use Eq. (3.6). This minimization problem is solved using Matlab function fminsearch which uses the simplex method.

min

C, K n

ÿ

i=1

(log(Ni)´log(Nexpi ))2 (3.6)

where Niis fatigue life predicted by the model and Nexpi is experimental fatigue life.

The calibrated material parameters are found to be σe = 491MPa, A = 0.1376, C = 0.7286 ,K

= 2.0032E-5 and L = 14.4. The fig. (3.5), (3.6) and (3.7) are Wöhler curves showing the initial parameters and calibrated parameters for the different mean stress cases. As the parameters are calibrated for each mean stress case, there is a small difference between the calibrated curves and the experimental data point as seen in fig. (3.8).

3.3. Calibration of Parameters

104 105 106 107

Life in Log scale 480 500 520 540 560 580 600 620 640 660 680 Stress Amplitude(MPa) Initial Parameters Experimental Data Calibrated Parameters

Figure 3.5: Wöhler Curve between initial and calibrated parameters at σm=0 MPa

104 ₁₀5 ₁₀6 ₁₀7

Life in Log scale 480 500 520 540 560 580 600 620 640 660 Stress Amplitude(MPa) Initial Parameters Experimental Data Calibrated Parameters

Figure 3.6: Wöhler Curve between initial and calibrated parameters at σm=100 MPa

104 105 106 107

Life in Log scale 480 500 520 540 560 580 600 620 640 Stress Amplitude(MPa) Initial Parameters Experimental data Calibrated Parameters

3.3. Calibration of Parameters 102 104 106 108 1010

N in cycles

S in MPa

4

Case Study and Results

4.1

Comparison with Cycle Counting Methods

In the conventional analysis of fatigue life of a component with an arbitrary load history, the cycle counting method is a key step in the analysis, as it defines the range and the mean values of the stress which contribute to fatigue damage. One scalar stress measure has to be defined for the analysis and this constitutes the basis of the counting algorithm, and there are many ways to choose this measure. These stress ranges and mean stresses are interpolated with the experimental data from the SN curves fig. (3.8) to obtain the fatigue life and total damage from, for example Palmgren-Miner rule as seen in Eq. (2.5). This is the standard methodology for fatigue life analysis. Therefore it is critical that the CTF model produces comparable results as to the standard methods.

Rainflow counting

There are various methods of cycle counting like peak counting, range counting, range mean counting etc. but the most widely used counting method is the rainflow counting [19]. Rain-flow counting is performed by taking the smallest repeating load cycles, inverting the plot by 90 degrees and considering stress peaks to be ’pagoda roofs’ and the droplet of water is drop-ping from this roof. The rules for the flow of this droplet can be obtained from [4]. From this method we obtain several stress ranges from which we can calculate the stress amplitudes and mean stresses of each half-cycle.

Methodology and Results

A tension/compression case is implemented for this analysis and as simple cyclic loading will produce the same results from both cycle counting and the CTF model, an undulating load is implemented. The undulating load is slightly more complex and hence allows for better comparison between the models. The undulating load cycle created with a particular overloading amplitude and frequency as seen in fig. (4.1) and fig. (4.2) is used in the study. The formulation for the undulating load is obtained by considering an overloading frequency (ωo) and overloading amplitude (Oa). The ωo is obtained by using a multiplier (m) on the

4.1. Comparison with Cycle Counting Methods 0 1 2 3 4 5 6 7 8 9 10 Time in s -600 -400 -200 0 200 400 600 Stress in MPa

Figure 4.1: Undulating Load Cycle

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Time in s -600 -400 -200 0 200 400 600 Stress in MPa

Figure 4.2: Zoomed in for clearer view Table 4.1: Cycle counting results for a small sample

Range Stress Range (MPa) Mean Stress (MPa)

1 41.81 -0.04274 1 244.73 0.004388 1 464.29 0.01482 1 668.09 0.222 1 849.40 0.1782 1 996.55 0.32 1 1108.52 0.1586 1 1172.87 0.4944 0.5 1186.46 6.764

here m is set to be 4 and this gives 16 peaks per cycle. As ω is set as 2π and the time period for each cycle is T=2π/ω, each cycle is 1 second long. Oais set as 1, and therefore the stress

amplitude remains the same and is not overloaded. The component for undulation (Of) is

given by

Of =1 ´ Oa+Oacos(ωot), ωo=2mω (4.1)

and the Undulating stress is given by

σ=σasin(ωt)Of(t) (4.2)

For the purpose of this work the function rainflow in MATLAB is used to obtain the various stress cycles and ranges [6]. From the data so obtained we see that the mean stresses for each stress cycles is much smaller than the stress amplitude, so we neglect the effect of the mean stress as seen in table 4.1, which shows values for a one small section. Also all values below Se are neglected as these will always give infinite fatigue life. The remaining stress

amplitudes are interpolated on a SN curve as seen in fig. (3.8) to obtain the fatigue life and applying Eq.(2.5) we obtain the Damage to be 8.5487E-5 s-1.

The same undulating load is then inserted into the CTF model and the fatigue life and damage development is obtained. The Eq.(3.5) is utilized to plot the endurance surface development as seen in fig.(4.3). We find the damage from this case to be 1.0595E-4 s-1. The difference between the rainflow analysis and CTF model is 19.27 %.

4.2. Validity Range for Non-proportional Loading 0 0.5 1 1.5 2 2.5 3 Time (s) -600 -400 -200 0 200 400 600 Stress (MPa)

Figure 4.3: Endurance Surface development for the undulating load (Zoomed in for clearer view)

4.2

Validity Range for Non-proportional Loading

Tension-Tension Load Combination

When a system is under multiaxial loading there can be non-proportionality between the var-ious applied loads. This requires the CTF model to produce accurate results for all such cases. By modelling the system to perform under various phase differences between the orthogonal stress fluctuations the reaction of the model can be analysed. The model is analysed for stress according to Eq. (4.3) for the phase differences of φ= (0, π/6, π/4, π/3, π/2, π)radians.

  σ11 0 0 0 σ22 0 0 0 0 

 and σ₁=σasinωt σ2=σasin(ωt+φ) (4.3)

The stress path for each value of φ, can be seen in fig. (4.4). The CTF model is implemented for each φ at a stress amplitude of σa = 400MPa to compute damage, as can be seen in fig.

(4.5). The damage for the stress state at φ = π radian is extremely high when compared to

the other phase differences.

This increase can be understood by considering a simplified case where effective stress σe f f

is computed from Eq. (4.3) for σa =1 and φ= 0 and πrespectively, also the back stress α=0. The effective stress for the out-of-phase loading is

?

3 times more than that of in-phase loading, this factor causes a sharp increase in the damage as seen in fig. (4.5). Therefore care must be taken to devise stress combinations that produce the same effective stress amplitude to highlight the dependence on phase angle.

σ=sinωt   1 0 0 0 1 0 0 0 0   and s=sinωt   1/3 0 0 0 1/3 0 0 0 ´2/3  , φ=0

4.2. Validity Range for Non-proportional Loading -400 -300 -200 -100 0 100 200 300 400 11 (MPa) -400 -300 -200 -100 0 100 200 300 400 22 (MPa) =0 = /6 = /4 = /3 = /2 =

Figure 4.4: Stress Path for Eq.(4.3)

0 1 2 3 4 5 6 7 8 9 10 Time (s) 10-15 10-10 10-5 100 Damage =0 = /6 = /4 = /3 = /2 =

Figure 4.5: Damage development for Eq.(4.3)

σe f f = c 3 2||s||= c 3 2 a s : s=sinωt Similarly, σ=sinωt   1 0 0 0 ´1 0 0 0 0  =s, φ=π σe f f = c 3 2||s||= c 3 2 a s : s=?3sinωt

Tension-Shear Load Combination

To completely study the effects of non-proportionality on the model it is important to have the same effective stress at both in-phase and out-of-phase loading condition. For this we consider a load combination of tension and shear with the stress as follows,

σ=   σ11 σ12 0 σ12 0 0 0 0 0   and σ₁₁= σa ? 2cosφ₂sinωt , σ12 = σa ? 6cosφ₂sin(ωt+φ) (4.4) As the shear stress component for the maximum von Mises stress is?3 times greater than that of the normal stress, the shear stress component for the load combination is reduced by a factor of?3 as compared to that of the normal stress. The pre-factor of these stress com-ponents is chosen such that the maximum von Mises stress fluctuation for in-phase loading is σa and for all out-of-phase loading the same maximum von Mises stress is present. This

combination of tension and shear load is implemented with the CTF model at various phase differences (φ = 0, π/6, π/4, π/3, π/2) and at σa = 600MPa and ω = 2π radians to

ob-tain the damage. The fig. (4.6) shows the damage development for the tension-shear load combination with same effective stress amplitude, when φ =π/2 the CTF model produces

unrealistic predictions. Therefore we can conclude that the model loses its validity at a non-proportional load which is rotary in nature as seen in fig. (4.7).

4.3. Validity Range of the Evolved Model for Non-proportional loading 0 50 100 150 200 250 300 Time (s) 0 1 2 3 4 5 6 7 8 9 10 Damage 10-3 =0 = /6 = /4 = /3 = /2

Figure 4.6: Damage development for Tension-shear Load for the CTF model

Figure 4.7: Stress path for combination of tension-shear load

4.3

Validity Range of the Evolved Model for Non-proportional loading

Implementation of Evolved Model with Tension-Shear Load Combination

The evolved model employs the equations from section 2.4 and implements them for the load combination of tension and shear. The stress model has same input parameters as in the case of the original model, Eq. (4.4). The ramp function used here R(u : v)as given by Eq.(2.27). The damage development is plotted as seen in fig.(4.8). The evolved model prediction are identical to the original model same as in fig. (4.6). The unrealistic predictions for φ=π/2 is

produced as the stress path for this phase difference is rotary as seen in fig. (4.7) and it causes the gradient (u) and loading directions (v) to become perpendicular as explained in section 2.4.

4.3. Validity Range of the Evolved Model for Non-proportional loading 0 50 100 150 200 250 300 Time (s) 0 1 2 3 4 5 6 7 8 9 10 Damage 10-3 =0 = /6 = /4 = /3 = /2

Figure 4.8: Damage development of the evolved model

Wöhler curves to identify effect of step a, in the ramp function S

To increase the validity range of the model the updated ramp function, S(u : v) given by Eq.(2.28) is used. Various values of a is implemented in the evolved model to obtain the damage evolution. The step size of a is set as a = (-0.005, -0.01, -0.015, -0.02).

We can see from fig. (4.9) that the damage steadily increases with increasing value of a, due to lack of experiments it becomes difficult to find the exact value of a that produces right damage. The Wöhler curve is plotted for the various values of a along with the curve at proportional loading to see the influence of a. From the fig. (4.10) we can see that a « ´0.01 produces a curve close to the proportional loading case, which is seen in the experiments. The slopes of these curves are not comparable with experimental results and they should follow the same pattern. The parameter L is used to change the slope.

Wöhler curves to study the effect of L

To calibrate the Wöhler curves from fig. (4.10) we run the model with various L. As L is set as a constant during the fitting procedure, it can be moved freely without compromising the goodness-of-fit. The effect of L on the fatigue life for non-proportional loading is studied by first calibrating the parameters C and K for the various L, and the result are given in the table 4.2 and plotted in fig.(4.11) and fig.(4.12).

Table 4.2: Calibrated parameters

L C K

1 14.4 0.7286 2.0032e-05 2 10 0.3304 2.1698e-05 3 20 1.8708 2.9076e-05

4.3. Validity Range of the Evolved Model for Non-proportional loading 0 50 100 150 200 250 300 Time (s) 0 0.005 0.01 0.015 0.02 0.025 0.03 Damage a=0 a=-0.005 a=-0.01 a=-0.015 a=-0.02

Figure 4.9: Damage development for various values of a at φ=π/2

102 103 104 105 106 107 Life in cycles 450 500 550 600 650 700 Stress in MPa a=0 a=-0.005 a=-0.01 a=-0.015 a=-0.02 a=0, =0

Figure 4.10: Wöhler Curve for various values of a

The parameters for each value of L is implemented in the evolved model with updated ramp function S(u : v) for non-proportional loading at φ = π/2 and a = ´0.01 to obtain the

Wöhler curve. These curves are plotted along with the proportional load case as seen in fig.(4.13). The slope of the curve increase with increasing L and this can be used to easily identify a suitable value when optimizing the solution.

4.3. Validity Range of the Evolved Model for Non-proportional loading

104 105 106 107

Life in Log scale 480 500 520 540 560 580 600 620 640 660 680 Stress Amplitude(MPa) Experimental Data Calibrated for L=14.4 Calibrated for L=10 Calibrated for L=20

Figure 4.11: Wöhler curve for the calibrated parameters σm=0

104 ₁₀5 ₁₀6 ₁₀7

Life in Log scale 480 500 520 540 560 580 600 620 640 660 Stress Amplitude(MPa) Experimental Data Calibrated for L=14.4 Calibrated for L=10 Calibrated for L=20 104 ₁₀5 ₁₀6 ₁₀7

Life in Log scale 460 480 500 520 540 560 580 600 620 640 Stress Amplitude(MPa) Experimental Data Calibrated for L=14.4 Calibrated for L=10 Calibrated for L=20

Figure 4.12: Wöhler curve for the calibrated parameters σm =100 MPa (left) and σm =200

MPa (right) 101 102 103 104 105 106 107 Life in cycles 450 500 550 600 650 700 Stress in MPa a=-0.01, L=14.4, = /2 a=-0.01, L=10, = /2 a=-0.01, L=20, = /2 a=0, L=14.4, =0

5

Discussion

5.1

Results

Comparison with Cycle Counting Methods

By comparing the result from the cycle counting method, the reliability of the CTF model can be studied. For a case of sinusoidal stress fluctuation the CTF model produces the same predictions as rainflow counting. However we can see from the results in section 4.1 for an undulating stress fluctuation the CTF model predicts 19.27 % higher cumulative damage than with the cycle counting method. The reason for this can be seen in fig. (4.3) where the endurance surface development is plotted, as seen here the damage begins to develop before the peak. Whereas the cycle counting methods only utilize the peak to measure the fatigue life and hence damage.

It is important to note that only rainflow counting was analyzed while other cycle counting methods are omitted. Further, the model uses an undulating load which becomes periodic over time, further analysis with a completely arbitrary load cycle might be used to study movement of the endurance surface and the fatigue life prediction. Also the experimental results might be different from rainflow counting, and therefore it also needs to be studied further.

Validity Range for Non-proportional loading

The initial tension-tension load combination predicts the increase in damage accurately for the case of φ = π, but we can see that the stress path for this case is linear as seen in fig.

(4.4). This leads to periodic loading in the model as mentioned in [13]. And due to the sharp contrast between the linear load case and the rotary load case (φ=π/2), it becomes evident

that a new load combination has to be implemented to study the effects in the rotary stress state at constant effective stress. In the tension-shear load combination, the accumulated damage at φ = π/2 is not in agreement with the expected fatigue behaviour, and it gives

much greater fatigue life than the in-phase loading and experiments show that in fact the fatigue life should be shorter for φ = π/2, [16, 1]. This failure to model the varied cases

5.2. Impact

Validity Range of the Evolved model for Non-proportional Loading

The evolved model with initial ramp function R, produces the same damage development curve as in the case of the original model as seen in fig. (4.6) and fig. (4.8). As the stress form a rotary path, the gradient and loading direction becomes perpendicular and the cosine of u : v, becomes 0. This implies no damage is developed at these points, which is not in agreement with experiments.

The updated ramp function S, steps the function before becoming 0, it mitigates this problem and corrects the damage development curve to be in qualitative agreement with experiments for φ=π/2.

The step a makes the ramp function allow for damage when u; v=0 and therefore increases the damage development. However, it is important to use a small step size to ramp. The effect of step a is seen in fig. (4.9). Here we can see that there is an increase in accumulated damage for each increment of the step. To identify the right step size Wöhler curve is plotted with the proportional load case for the various values of a. For a = -0.01, a curve closest to the proportional load case is obtained. The effect of the material parameters on the Wöhler curve is already studied in the calibration. As the slope of the curve a = ´0.01 is not the same as the proportional load case, these parameters can be varied and fig. (4.13) shows that by tuning the parameter L, there is qualitative improvement in the fatigue behaviour for

φ=π/2.

Further, experimental study of the material for non-proportional loading for π/2 phase difference needs to be performed so that there will be sufficient data to calibrate the step a in the ramp function S. The instability caused by the changes in L also has to be studied further.

5.2

Impact

The CTF model has application in high fidelity fatigue analysis wherein the damage can be simultaneously computed along with the stresses. For models which have billions of nodes, this mitigates the requirement for data storage as fatigue analysis is generally performed in post-processing. This will reduce the simulation time and hence provide economical benefits. The evolved model increases the analytical capability in the non-proportional load cases. The CTF model will be particularly advantageous in the aircraft or automotive application where the entire structure is manufactured from individual blocks of material. The CTF will be able to analyse these complex structures in its entirety and will reduce the need for part simplification. Further the model can handle complex load histories and thus real life load conditions can be implemented on the system for analysis. A comprehensive study on the fatigue behaviour of a component provides a good metric on time for failure and hence enable safe application.

6

Conclusion

In this thesis, the CTF model is compared with the cycle counting method and it is found that the model predicts the fatigue behaviour of the material well within reason. It can even be argued that the CTF model predicts the damage development closer to reality than cycle counting, as each point of the cycle is computed for damage rather than just the peak. The non-proportional behaviour of the model is studied and this shows the CTF model in its original formulation fails at rotary stress states. The proposed evolved continuous time fatigue model improves the fatigue behaviour of the original model by implementing a geometric framework that particularly addresses issues with non-proportional loading. The results so obtained predict correct fatigue behaviour but it contains instabilities that need to be further studied. The CTF model provides a wide range of advantages for fatigue analysis, so a comprehensive development of this will be beneficial as a tool for fatigue analysis in in-dustries. A new parameter fitting procedure will have to be developed such that it considers the extended parameter set; this will require new experiments for non-proportional loading. Effects of temperature on the fatigue behavior should also be considered for the CTF model. Further, anisotropical behaviour which is present during the crack initiation phase should also be studied.

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