Mechanical behaviour of AZ31B Magnesium alloy subjected to in … · iii Abstract The present work was carried out in order to better understand and characterize the mechanical behaviour

Mechanical behaviour of AZ31B Magnesium alloy

subjected to in-plane biaxial fatigue

Ricardo José Almeida Silva Pereira

Thesis to obtain the Master of Science Degree in

Mechanical Engineering

Supervisors: Prof. Luís Filipe Galrão dos Reis

Prof. Ricardo Miguel Gomes Simões Baptista

Examination Committee

Chairperson: Prof. João Orlando Marques Gameiro Folgado

Supervisor: Prof. Luís Filipe Galrão dos Reis

Members of the Committee: Prof. Rui Fernando dos Santos Pereira Martins

May 2016

To my parents, my sister and my niece

i

Acknowledgements

This work would not have been possible without the many contributions, from various

people, therefore I would like to show my appreciation to the following:

First of all, I would like to show my gratitude and appreciation to professor Luís Reis, for

presenting me the opportunity to develop this work under his supervision and also for the

constant support and incentive to keep pushing through.

Professor Ricardo Baptista for sharing his knowledge especially regarding crack

modelling in finite elements.

Professor Ricardo Cláudio for taking the time to suggest solutions and share his

knowledge with the biaxial testing machine.

Professor Mafalda Guedes for all her input regarding surface polishing.

Professor Carlos Fortes for providing the CNC programming to manufacture the

specimens.

Gonçalo Torres, lab technician who always had availability to help with tools or

assembling of specimens.

Cátia Piedade, for the company and the input regarding surface polishing.

Márcio Farinhas for always being available in the CNC workshop.

Tiago Marques for the contribution with the fracture surface photographs.

To my closest friends for the constant support.

To my family, for everything.

ii

Resumo

O presente trabalho foi desenvolvido com o intuito de caracterizar e compreender o

comportamento mecânico da liga de magnésio AZ31B quando sujeita a fadiga multiaxial. O

estudo foi desenvolvido com recurso a ensaios experimentais realizados com provetes

cruciformes, obtidos de chapa com 3.25 mm de espessura, e cuja geometria foi optimizada

para estes testes em que o carregamento é biaxial no plano. A máquina de ensaios foi

desenvolvida internamente e foi construída com recurso a 4 motores lineares e com um

sistema de guiamento não-convencional que permite testar materiais de engenharia de forma

precisa e eficiente.

Os ensaios experimentais foram realizados com carregamentos sinusoidais tanto para

casos em fase como desfasados, com rácio constante e tensão média igual a zero. A

monitorização da iniciação e propagação de fendas foi efectuada através de fotografias obtidas

por um microscópio USB, em intervalos pré-definidos por um determinado número de ciclos.

Os resultados dos modelos de plano crítico apresentaram uma boa relação com os

modelos que definem o plano crítico com base nas tensões e/ou extensões normais. No caso

da propagação, as estimativas obtidas por análises de elementos finitos devolveram resultados

coerentes exceptuando o caso do carregamento desfasado de 180° quando relacionados com

os dados experimentais de propagação da fenda. Ao longo dos vários ensaios, a iniciação e

propagação demonstraram a tendência para ocorrer em direcções aproximadamente

perpendiculares à direcção de laminagem.

Palavras-chave:

Fadiga biaxial, liga de Magnésio, ensaios experimentais, provetes cruciformes.

iii

Abstract

The present work was carried out in order to better understand and characterize the

mechanical behaviour of the magnesium alloy AZ31B, subjected to multiaxial fatigue. The study

was conducted by performing experimental tests on cruciform specimens with a geometry

specially optimized for use in these tests, obtained from 3.25 mm thick sheet, subjected to in-

plane biaxial loading. The testing apparatus used was a biaxial testing machine developed in-

house and built with four iron-core linear motors and with a non-conventional guiding device

which allows for precise and efficient experimental testing of engineering materials.

The tests were performed with sinewave loadings for both in-phase and out-of-phase

cases, with constant load ratio and mean stress equal to zero. With crack initiation and

propagation being monitored recurring to a USB microscope that took snapshots on periodic

intervals defined by the number of cycles.

The critical plane results were reasonably accurate for models that defined the critical

plane based only on normal stresses and/or strains. For crack propagation, the estimations

obtained from finite element analyses provided reasonable results except for the case of the

fully reversed loading path when related with the experimental data regarding crack

propagation. Throughout all tests, crack initiation and propagation showed a trend to occur in

directions approximately normal to the rolling direction.

Keywords:

Biaxial fatigue, Magnesium alloy, experimental tests, cruciform specimens.

iv

Table of Contents

Acknowledgements ......................................................................................................... i

Resumo ............................................................................................................................ ii

Palavras-chave: .............................................................................................................. ii

Abstract .......................................................................................................................... iii

Keywords: ...................................................................................................................... iii

Table of Contents .......................................................................................................... iv

List of Figures ............................................................................................................... vii

List of Tables................................................................................................................... x

List of Acronyms ........................................................................................................... xi

List of Symbols ............................................................................................................. xii

1 Introduction .............................................................................................................. 1

1.1 Motivation and Framework ................................................................................ 1

1.2 Objectives .......................................................................................................... 2

1.3 Thesis Structure ................................................................................................ 2

2 Bibliographical Review ........................................................................................... 4

2.1 Magnesium ........................................................................................................ 4

2.1.1 Magnesium metal production ........................................................................ 4

2.1.2 Magnesium alloys and crystal structure ........................................................ 5

2.1.3 Manufacturing processes used for magnesium alloy components ............... 6

2.1.4 Applications of magnesium alloys ................................................................. 8

2.1.4.1 Aerospace industry .................................................................................... 8

2.1.4.2 Automotive industry ................................................................................... 9

2.1.4.3 Other applications.................................................................................... 11

2.2 Fatigue ............................................................................................................. 11

2.2.1 Historical summary ...................................................................................... 11

2.2.2 Multiaxial Fatigue ........................................................................................ 18

2.2.2.1 Proportional Loading ............................................................................... 18

2.2.2.2 Nonproportional loading .......................................................................... 19

2.2.3 Material Behaviour ...................................................................................... 20

v

2.2.3.1 Isotropic hardening .................................................................................. 21

2.2.3.2 Kinematic Hardening ............................................................................... 22

2.2.3.3 Cyclic Creep or Ratcheting ...................................................................... 22

2.2.3.4 Mean stress relaxation ............................................................................ 23

2.2.3.5 Nonproportional cyclic hardening ............................................................ 24

2.2.4 Fatigue Crack Growth ................................................................................. 24

2.2.5 Fatigue Life .................................................................................................. 27

2.2.6 Design Theories .......................................................................................... 27

2.2.6.1 Infinite-Life Design ................................................................................... 27

2.2.6.2 Safe-Life Design ...................................................................................... 27

2.2.6.3 Fail-Safe Design ...................................................................................... 27

2.2.6.4 Damage-Tolerant Design ........................................................................ 28

2.2.7 Fatigue Models ............................................................................................ 28

2.2.7.1 Findley ..................................................................................................... 28

2.2.7.2 Brown and Miller ...................................................................................... 29

2.2.7.3 Fatemi and Socie..................................................................................... 30

2.2.7.4 Smith, Watson and Topper ...................................................................... 31

2.2.7.5 Liu I and Liu II .......................................................................................... 32

2.2.7.6 Chu, Conle and Bonnen .......................................................................... 33

3 Material, Equipment and Methods ....................................................................... 34

3.1 Material ............................................................................................................ 34

3.2 Specimen Geometry ........................................................................................ 35

3.3 Equipment ....................................................................................................... 39

3.3.1 Testing apparatus ........................................................................................ 39

3.3.2 USB Microscope .......................................................................................... 40

3.4 Experimental Methods ..................................................................................... 40

4 Numeric Study ....................................................................................................... 43

4.1 Specimen and crack modelling ....................................................................... 43

4.2 Mesh and element type ................................................................................... 45

4.3 Boundary Conditions and Loads ..................................................................... 46

4.4 Theoretical concepts applied to the numeric study ......................................... 47

vi

5 Results and Discussion ........................................................................................ 48

5.1 Critical plane models ....................................................................................... 48

5.1.1 Findley Model .............................................................................................. 48

5.1.2 Brown and Miller Model ............................................................................... 49

5.1.3 Fatemi and Socie Model ............................................................................. 49

5.1.4 Smith, Watson and Topper Model ............................................................... 50

5.1.5 Liu I and II Model ......................................................................................... 51

5.1.6 Chu, Conle and Bonnen Model ................................................................... 52

5.2 Experimental results ........................................................................................ 53

5.2.1 Crack Initiation ............................................................................................. 53

5.2.2 Crack Propagation ....................................................................................... 55

5.3 Numeric study results ...................................................................................... 61

5.3.1 Specimen 004 and 005 ............................................................................... 61

5.3.2 Specimen 008 ............................................................................................. 63

5.3.3 Specimen 009 ............................................................................................. 64

5.3.4 Specimen 010 ............................................................................................. 65

5.4 Correlation of experimental and numeric data ................................................ 66

5.5 Fracture surface analysis ................................................................................ 69

6 Conclusions and Future Developments.............................................................. 72

6.1 Conclusions ..................................................................................................... 72

6.2 Future Developments ...................................................................................... 72

References .................................................................................................................... 73

vii

List of Figures

Figure 1.1 – Infamous fatigue failures of the 20th century (a) Alexander L. Kielland platform, [2];

(b) Aloha Airlines Flight 243, [3]. ................................................................................................... 1

Figure 2.1 – Layering of a Hexagonal close-packed structure, [12].............................................. 6

Figure 2.2 – Military aircraft applications that employ Magnesium (a) Sikorsky S-56 [17]; (b)

Lockheed F-80C [18]; (c) Convair B-36 Peacemaker [19]; (d) Tupolev TU-95MS [20] ................ 8

Figure 2.3 – Automotive application of magnesium alloys (a) 1938 VW Beetle [24]; (b) Allard

sports car [5]; (c) Mercedes-Benz 300 SLR [5] ............................................................................. 9

Figure 2.4 – Drivetrain applications of Mg alloys (a) Mercedes 7-speed automatic transmission

housing [7]; (b) Audi V8 intake manifold [5]; (c) Mercedes M291 prototype crankcase [5]; (d)

BMW cylinder head cover [5]; (e) Mercedes M291 prototype engine block [5] .......................... 10

Figure 2.5 – Automotive applications (a) Steering wheel armature [5]; (b) Seat frame [5]; (c)

Inner door panel [25]; (d) Tailgate [5]; (e) Bonnet inner part [5] .................................................. 10

Figure 2.6 – Proportional multiaxial loading, [29]. ....................................................................... 19

Figure 2.7 – Nonproportional loading, [29]. ................................................................................. 19

Figure 2.8 – Intrusion-Extrusion model that leads to slip band formation, [29]. .......................... 20

Figure 2.9 – Isotropic Hardening, [29]. ........................................................................................ 21

Figure 2.10 – Kinematic hardening, [29]. .................................................................................... 22

Figure 2.11 – Ratcheting, [29]. .................................................................................................... 23

Figure 2.12 – Mean stress relaxation, [29]. ................................................................................. 23

Figure 2.13 – Cyclic stress-strain curve for proportional and nonproportional loading, [29]. ...... 24

Figure 2.14 – Different modes of crack loading, [63]. ................................................................. 25

Figure 2.15 – Stage I and II crack growth, [29]. .......................................................................... 25

Figure 2.16 – Relation between 𝑑𝑎/𝑑𝑁 and ∆𝐾, adapted from [64]. .......................................... 26

Figure 2.17 – Schematic representation of fatigue life, [65]. ....................................................... 27

Figure 2.18 – Schematic representation of the Damage-Tolerant Design concept, [66]. ........... 28

Figure 2.19 – (a) Case A; (b) Case B cracks, [29]. ..................................................................... 29

Figure 2.20 – Fatemi and Socie’s model schematic illustration, [29]. ......................................... 31

Figure 2.21 - Smith, Watson and Topper crack growth mechanism, [29]. ................................. 31

Figure 2.22 - Elastic and plastic strain energies, [67]. ............................................................... 32

Figure 3.1 – General geometry of the cruciform specimen ......................................................... 35

Figure 3.2 – Specimen geometry: (a) after the first stage; (b) after the second stage. .............. 37

Figure 3.3 – Biaxial Testing Machine used to perform the experimental tests, [69]. .................. 39

Figure 3.4 – Driving system assembly, [69]. ............................................................................... 39

Figure 3.5 – Variation of loads during a complete cycle: (a) In-phase loading; (b) phase shift of

45°; (c) phase shift of 90°; (d) phase shift of 180°. ..................................................................... 41

Figure 3.6 – Resulting load paths: (a) In-phase loading; (b) phase shift of 45°; (c) phase shift of

90°; (d) phase shift of 180°. ......................................................................................................... 42

viii

Figure 4.1 – Specimen model in ABAQUS®. ............................................................................... 43

Figure 4.2 – Ideal and real crack tip comparison, [63]. ............................................................... 43

Figure 4.3 – Crack tip model detail.............................................................................................. 44

Figure 4.4 – Close-up of both cracks modelled in ABAQUS®. .................................................... 44

Figure 4.5 – Element types used for the specimen model; (a) C3D15 wedge element; (b)

C3D20R brick element, [71]. ....................................................................................................... 45

Figure 4.6 – Specimen mesh (a) Crack tip detail; (b) Sample of the rest of the mesh. .............. 45

Figure 4.7 – Displacement boundary conditions on the specimens extremities. (a) along the x

direction; (b) along the y direction. .............................................................................................. 46

Figure 4.8 – Loads applied on the specimens extremities. (a) along the x direction; (b) along the

y direction. ................................................................................................................................... 47

Figure 5.1 – Findley parameter variation. ................................................................................... 48

Figure 5.2 – Brown and Miller parameter variation. .................................................................... 49

Figure 5.3 – Fatemi and Socie parameter variation. ................................................................... 50

Figure 5.4 – Smith, Watson and Topper parameter variation. .................................................... 50

Figure 5.5 – Liu I parameter variation. ........................................................................................ 51

Figure 5.6 – Liu II parameter variation. ....................................................................................... 52

Figure 5.7 – Chu, Conle and Bonnen parameter variation. ........................................................ 52

Figure 5.8 – Specimen BTM2022-004; (a) at 453924 cycles; (b) at 456444 cycles. .................. 53

Figure 5.9 – Specimen BTM2022-008; (a) at 38905 cycles; (b) at 39927 cycles. ...................... 54

Figure 5.10 – Specimen BTM2022-009; (a) at 567495 cycles; (b) at 568519 cycles. ................ 54

Figure 5.11 – Specimen BTM2022-010; (a) at 980140 cycles; (b) at 981162 cycles. ................ 54

Figure 5.12 – Crack length vs number of cycles for specimen 004. ........................................... 55

Figure 5.13 – Crack propagation of specimen 004; (a) at 458964 cycles; (b) at 461484 cycles;

(c) at 479117 cycles; (d) at final fracture (480826 cycles). ......................................................... 56


Figure 5.15 – Crack propagation of specimen 005; (a) at 44995 cycles; (b) at 47015 cycles; (c)

at 53072 cycles; (d) at final fracture (60091 cycles).................................................................... 57


Figure 5.17 – Crack propagation of specimen 008; (a) at 41959 cycles; (b) at 43999 cycles; (c)

at 52156 cycles; (d) at 62332 cycles. .......................................................................................... 58



(c) at 581804 cycles; (d) at final fracture (590701 cycles). ......................................................... 60



(c) at 1025152 cycles; (d) at final fracture (1029244 cycles). ..................................................... 61

Figure 5.22 – Stress distribution at crack tip for specimen 004. ................................................. 62



ix



Figure 5.27 – da/dN vs ΔKeq ........................................................................................................ 67




Figure 5.31 – Fracture surfaces of specimen 003....................................................................... 70



x

List of Tables

Table 2.1 – Physical properties of pure Magnesium [9] ................................................................ 5

Table 3.1 – AZ31B-H24 properties [68] ...................................................................................... 34

Table 3.2 – Percentage range of the alloying elements in the AZ31B-H24 alloy [68] ................ 34

Table 3.3 – Values of the design variables considered .............................................................. 36

Table 3.4 – Operation sequence, type of tool and cutting parameters ....................................... 37

Table 3.5 – Measured values of centre thickness. ...................................................................... 38

Table 3.6 – BTM specifications ................................................................................................... 40

Table 3.7 – Test parameters for each specimen ......................................................................... 41

Table 5.1 – Comparative overview of theoretical and experimental results for crack initiation. . 55

Table 5.2 – Equivalent SIF range relation with respective half crack length for specimen 004 .. 62



Table 5.5 – Equivalent SIF range relation with respective half crack length for specimen 009. . 65

Table 5.6 – Equivalent SIF range relation with respective half crack length for specimen 010. . 66

xi

List of Acronyms

AISI - American Iron and Steel Institute

ASME - American Society of Mechanical Engineers

ASTM - American Society for Testing and Materials

BCC - Body Centred Cubic

BTM - Biaxial Testing Machine

CNC - Computerized Numeric Control

FEA - Finite Element Analysis

FS - Fatemi and Socie

HCF - High-Cycle Fatigue

HCP - Hexagonal Close-Packed

IPS - Instituto Politécnico de Setúbal

IST - Instituto Superior Técnico

LCF - Low-Cycle Fatigue

LEFM - Linear Elastic Fracture Mechanics

MCC - Minimum Circumscribed Circle

MCE - Minimum Circumscribed Ellipse

SIF - Stress Intensity Factor

SSF - Stress Scale Factor

SSM - Semi-Solid Metal

SWT - Smith, Watson and Topper

UAV - Unmanned Aerial Vehicle

VSE - Virtual Strain Energy

xii

List of Symbols

Greek notation

∆𝛾 – Shear strain range

∆𝛾 – Equivalent shear strain range

𝛥𝛾𝑚𝑎𝑥 – Maximum shear strain range

𝛿 – Phase shift

∆휀𝑛 - Normal strain range

∆휀1 – Principal strain range

휀𝑛 – Normal strain

∆𝜎 – Normal stress range

𝜎𝑎,𝑅=−1 – Alternate normal stress for a stress ratio of -1

𝜎𝑎 – Alternate normal stress

𝜎𝑛,𝑚𝑎𝑥 – Maximum normal stress

𝜎𝑛 – Normal stress

𝜎𝑦 – Material’s yield stress

𝛥𝜏 – Shear stress range

𝜏𝑎,𝑅=−1 – Alternate shear stress for a stress ratio of -1

𝜏𝑎 – Alternate shear stress

𝜏𝑛,max - Maximum Shear stress

𝜔 – Frequency

Roman notation

ΔK – Stress intensity factor range

∆𝐾𝑒𝑞 – Equivalent stress intensity factor range

∆𝑊𝐼 – Work quantity related with mode I

∆𝑊𝐼𝐼 – Work quantity related with mode II

xiii

𝐹1 – Load along direction 1

𝐹2 – Load along direction 2

𝐹𝑎 – Load amplitude

𝐾𝐼 – Stress intensity factor for mode I

𝐾𝐼𝐼 – Stress intensity factor for mode II

𝐾𝑒𝑞,𝑚𝑎𝑥 – Maximum equivalent stress intensity factor

𝐾𝑒𝑞,𝑚𝑖𝑛 – Minimum equivalent stress intensity factor

𝐾𝑒𝑞 – Equivalent stress intensity factor for mixed mode I-II

∆𝑊 – Virtual strain energy

a – Crack half-length at the surface of the specimen

C – Material parameter for Paris Law

da/dN – crack propagation rate

f – Findley’s damage parameter

k – Material parameter for the fatigue models

m – Paris law exponent

n – Basquin law exponent

N – Number of Cycles

S – Material parameter

t – time

Y – Shape factor

In case of specific symbols, the designation can be found in the text where it is referred.

1

1 Introduction

In this chapter a brief overview of the topics discussed in this thesis is presented, including the

motivation and framework, as well as the objective of the work and the structure of this document.

1.1 Motivation and Framework

Fatigue failure of mechanical components, structures and systems has been observed since

the 19th century, and has become a well-documented phenomenon to the present day. Simply put,

fatigue is a phenomenon due to the accumulation of damage, caused by cyclic loads.

Although no official figure is available, many sources suggest that 50 to 90 percent of all

mechanical failures are caused by fatigue, and most of these failures are unforeseen. The

considerably large percentage of failures due to fatigue, takes into account a wide range of

applications, from household items, such as door springs or tooth brushes, to much more complex

structures and systems, like ground vehicles, aircrafts or ships to name a few, [1]. A couple of

infamous fatigue failures are shown in Figure 1.1.

Figure 1.1 – Infamous fatigue failures of the 20th century (a) Alexander L. Kielland platform, [2]; (b)

Aloha Airlines Flight 243, [3].

The Norwegian platform shown in Figure 1.1a, claimed 123 lives in 1988 and it was caused by

a fatigue crack on a steel brace, [2]. The Aloha Airlines Flight 243 of Figure 1.1b happened in 1988

and part of the fuselage fractured in-flight due to fatigue crack which was caused by corroded rivet

holes, [3].

Since failure by fatigue impacts such a wide range of applications, it is engineering’s duty to

avoid this kind of failure, as it may carry dire consequences. Fatigue failures have claimed human

lives in some cases and generally carry a significant economic impact. Due to all of this, it is of the

utmost importance to study and understand fatigue in order to avoid the catastrophic failure of

structures. The present work presents an innovative study conducted with a testing apparatus

developed specifically to explore the in-plane biaxial fatigue loading conditions, and therefore attain

knowledge in the aforementioned loading conditions.

2

Magnesium alloys have become more and more desirable in recent times, mainly due to some

of its properties, such as its density for instance. The fact that magnesium alloys are the lightest alloys

available makes them a strong candidate to be used in several industries, specifically: automotive and

aerospace. In the aforementioned industries, the use of magnesium alloys tracks back to the late

1930’s and its use by automobile manufacturer Volkswagen, or Sikorsky helicopters in the 1950’s, and

extends to the present day, with magnesium alloys being used in high-end applications like Formula 1

and current aircraft models from Boeing. Despite the fact that the low density of magnesium alloys

provides advantages, there are also disadvantages related to these alloys, in particular Magnesium

alloys are prone to corrosion, and under certain conditions these alloys may present a fire hazard, due

to magnesium being a very reactive element.

Many strides have been made by research and development (R&D) departments and

universities regarding magnesium alloys in order to shed light on the capabilities of the alloys and to

broaden the range of application for the alloys.

Considering the topics previously stated, the main motivation behind this work lies on the

possibility of performing experimental tests, analyse the obtained data and provide a contribution of

knowledge to the scientific community, in a matter of high importance such as fatigue, with such an

appealing material as is the magnesium alloy discussed in this thesis.

1.2 Objectives

The main scope of this work is to perform experimental tests of magnesium alloy AZ31B-H24,

subjected to in-plane biaxial fatigue, with a specimen geometry optimized to study crack initiation and

propagation and correlate the experimental data with theoretical models. This correlation is intended

to be attained by comparing the crack initiation angles from the experimental tests, with the results of

the critical plane models; obtaining the stress intensity factor range from the numeric study as a

function of the crack size obtained from the experimental tests and relate it with the crack growth rate.

1.3 Thesis Structure

This thesis is composed of six chapters to which the contents are spread in the following

order.

Chapter 1 is an introductory chapter that provides the framework and the motivation behind

this work along with the objectives and the structure of the thesis.

Chapter 2 is dedicated to the literature review which includes a basic explanation on the

production process of magnesium from ore; magnesium alloy nomenclature and a brief explanation of

its crystal structure; identification of the production methods employed to manufacture magnesium

parts; applications of magnesium alloys; a historical summary relevant to the work developed and the

explanation of the theoretical concepts that support the study developed with this thesis.

3

Chapter 3 presents the overview of the material used in this study, a detailed explanation

regarding the specimen geometry and a brief overview of the manufacturing processes involved, a

description of the apparatus used in the experimental tests and the methodology to conduct the

experimental tests.

Chapter 4 shows the concepts related to the numerical study and the steps taken in this

analysis performed with the commercial finite element code ABAQUS®. The theoretical concepts

related to the treatment of the results obtained in this chapter, are also overviewed.

Chapter 5 consists of the presentation of the results obtained for the various approaches and

the respective discussion.

Chapter 6 presents the conclusions drawn from this study as well as some proposals for future

development.

4

2 Bibliographical Review

This chapter presents a bibliographical review that covers several aspects related to both

magnesium alloys and fatigue. It presents a brief explanation on magnesium metal production as well

as alloy nomenclature, crystal structure, component manufacturing processes and applications.

Regarding the topic of fatigue, a historical summary is presented while the remaining of the chapter is

devoted to explaining fundamental concepts regarding fatigue which are key to present a theoretical

background to the work performed.

2.1 Magnesium

2.1.1 Magnesium metal production

Magnesium is the eighth most common element on earth, constituting about 2% of the earth’s

crust. Magnesium can be found in mineral form and also dissolved in seawater, averaging a

concentration of 0.13%, indicating that magnesium is a resource close to being inexhaustible, [4], [5],

[6] and [7].

Production of magnesium metal (the product with highest interest for the present work) is

usually performed through one of two paths: it is either done by a thermal reduction process or by an

electrolytic process. However, there are variations depending on the manufacturer and also depending

on the actual method within the process. A third way of producing magnesium, is by means of

recycling, [4] and [8].

In the thermal reduction process, dolomite ore (a mineral composed of calcium magnesium

carbonate) is crushed and put in a thermally insulated chamber, designated kiln, in order for the

mineral to go through a process called calcining, which produces a mixture of magnesium and calcium

oxides. After obtaining the oxides, the magnesium oxide should be reduced, for the reduction to take

place, ferrosilicon is used. Ferrosilicon is then crushed and mixed with the oxides, and finally made

into briquettes that are loaded into a reactor. The reaction takes place under low pressure and in a

temperature range around 1200 to 1500 °C. The conditions mentioned produce magnesium as a

vapour, which is condensed by cooling to about 850 °C in steel-lined condensers, and afterwards

removed and cast into ingots.

For the electrolytic process two stages are required, first pure magnesium chloride should be

produced from seawater or brine and only then, can the electrolysis of fused magnesium chloride take

place. If magnesium chloride is produced from seawater, it must be treated with mixed oxides

(obtained from dolomite), inducing the precipitation of magnesium hydroxide, which when heated will

form magnesium oxide. To obtain magnesium chloride, the oxide should be heated while mixed with

carbon, in a stream of chlorine at high temperature in an electric furnace. Obtaining magnesium

chloride from brines requires evaporation stages to remove impurities. The product of the evaporation

stages has to go through a final stage of dehydration, which requires hydrogen chloride to be present

in gaseous form, in order to avoid hydrolysis of the magnesium chloride. Finally the magnesium

5

chloride obtained is subjected to electrolysis, where it is continuously fed into electrolytic cells, which

in turn are at temperatures high enough to melt it. This operation produces magnesium and chlorine.

The molten magnesium is then removed and cast into ingots.

2.1.2 Magnesium alloys and crystal structure

Magnesium, as a metal obtained through the methods stated previously, is not suitable for

mechanical applications. However, when alloyed to other elements, its properties improve

significantly.

Magnesium is the lightest structural metal available, with a density lower than aluminium’s by

about a third, and close to that of fibre reinforced plastics. The physical properties of magnesium are

shown in Table 2.1, [9].

Table 2.1 – Physical properties of pure Magnesium [9]

Property Value

Atomic number 12

Density when solid (at 20°C), [g cm-3

] 1.74

Density when liquid (at 651°C), [g cm-3

] 1.59

Melting point, [°C] 649

Boiling point, [°C] 1090

Thermal conductivity (at 0-100°C), [W (m K)-1

] 155.5

Specific heat (at 20°C), [J (kg K)-1

] 1022

Coefficient of thermal expansion (at 0-100°C), [10-6

K-1

] 26.0

Electrical resistivity (at 20°C), [µΩ] 4.2

Temperature coefficient of resistivity (at 0-100°C), [10-3

K-1

] 4.25

Magnesium is often alloyed with other elements, in order to improve its properties and become

suitable for a wider range of applications in several industries. The type of magnesium alloy mentioned

is easily identified by its designation. These designations are usually comprised by two letters,

followed by two numbers, and when applicable a third letter, and/or a fourth part consisting of a letter

followed by a number, separated from the third part of the designation by a hyphen. The first two

letters indicate the two main alloying elements, arranged in order of decreasing percentage, or

alphabetically in case the percentages are equal. The two numbers that follow, express the

percentages of the two main alloying elements. The letter in the third part distinguishes alloys with

slightly different compositions within the same main designation. The fourth part of the designation

indicates that the alloy has undergone some treatment, [10] and [11].

The correspondence between the letters used in the first part of the designation is as follows:

A – Aluminium; B – Bismuth; C – Copper; D – Cadmium; E – Rare Earth Elements; F – Iron; H –

Thorium; K – Zirconium; L – Lithium; M – Manganese; N – Nickel; P – Lead; Q – Silver, R –

6

Chromium; S – Silicon; T – Tin; W – Yttrium; Y – Antimony; Z – Zinc. Regarding the fourth part of the

designation, the code used has the following correspondence: F – As fabricated; O – As annealed;

H10 and H11 – Slightly strain hardened; H23, H24 and H26 – Strain hardened and partially annealed;

T4 – Solution heat treated; T5 – Artificially aged only; T6 – Solution heat treated and artificially aged;

T8 – Solution heat treated, cold worked and artificially aged, [11].

Magnesium has a hexagonal close-packed (HCP) crystal structure. The packing factor, which

indicates the portion of volume in a crystal structure that is occupied by the atoms that constitute said

volume, is 0.74. The layering of this type of crystal structure alternates between two equivalent shifted

positions, arranged in an ABAB sequence as shown in Figure 2.1, [12].

Figure 2.1 – Layering of a Hexagonal close-packed structure, [12].

This structure has implications regarding the behaviour of the material, since it is a non-cubic

lattice and that means slippage does not occur easily, which in turn makes the material less

deformable at room temperature, however, it can be deformed by conventional methods at higher

temperatures (in a range about 200 to 225 °C). At room temperatures, the only deformation

mechanisms are gliding and twinning.

2.1.3 Manufacturing processes used for magnesium alloy components

Although magnesium alloys’ mechanical properties are currently slightly lower than its main

competitors, they are still widely used in a variety of industries. For such uses, the alloys are mainly

divided into two categories: casting alloys and wrought alloys. Casting alloys can be manufactured into

magnesium alloy components through some conventional casting methods, [5].

Sand casting can be used to manufacture components without much change in usual

practices related to this process, there are however, a few particularities related to the characteristics

of magnesium (physical and chemical) that should be investigated in order to produce a given part

through this process. Die casting has similarities to the plastic injection moulding process and is

commonly used for high production rates. This process can achieve high dimensional accuracy,

produce parts with thin walls and improve productivity.

7

Another process that can be used is designated squeeze casting, which combines forging and

casting processes. This process can be defined as direct or indirect depending on the method used to

produce the actual part. In direct squeeze casting, molten magnesium is poured into a die at slow

speeds, once the die cavity is filled a punch is brought down, applying pressure until the metal

solidifies. In indirect squeeze casting the molten magnesium is poured into an encasement, after that,

the speed of the molten metal flowing into the mould is controlled by a plunger, [5].

Production of components by casting can also be done by means of Semi-Solid Metal (SSM)

casting. Within SSM casting, there is one method that is more commonly used with magnesium alloys,

this method is called Thixomolding and it was introduced in 1990 by Dow engineers. Thixomolding is

quite alike plastic injection moulding, only in thixomolding, the feeder is filled with magnesium chips,

taking the chips into a heated screw that starts heating the chips by rotating, while pushing them

simultaneously. The heat and shear forces produced by the screw, generate a semi-solid slurry, which

is injected into the mould to obtain the desired component [5].

Components made from magnesium alloys may also be obtained from wrought products like,

extrusions, forgings, sheet and plate. Regardless of way a component has been produced, there is

always room to give shape by means of machining.

Magnesium is a material with high machinability [13], this implies that the relative power

required for a certain operation is lower than for other metals. However, there is a drawback in the

midst of these characteristics, specifically, machining magnesium might present a fire hazard.

The machining operations can be performed by conventional manually-operated machine

tools, or purpose-built, automatized machine tools [13]. The fact that magnesium alloys have good

machinability allows for heavy cuts at high cutting speeds and feeds, which implies reduced operating

times. Besides the previously stated, high thermal conductivity and low cutting pressure let the

generated heat dissipate quickly, and thus improve tool life. The tools used in machining operations on

magnesium alloys, should be chosen with great care, generally, regular carbon steel tools, can be

used with satisfactory service lives. However, carbide-tip or diamond-tip tools can be used as well,

especially if very fine finishes are required [13]. Independently of the tool, these should be kept sharp

and smooth at all times to avoid poor surface finish, excessive heat, formation of long chips with

burnished surfaces and the occurrence of flashing or sparking at the tool edge.

There are certain characteristics that the tools used for machining magnesium alloys should

possess, such as large peripheral relief angles, large chip spaces, few blades (for certain milling

cutters) and small rake angles. Large relief and clearance angles are important in order to avoid

excessive heating [13].

Machining of magnesium alloys is often done without cutting fluids, due to the material’s

thermal conductivity and resistance to galling, cooling and lubrication are seldom needed [13].

Although magnesium is mostly machined without recurring to cutting fluids, in certain cases, usage of

said fluids might be required, particularly in operations that combine very high feeds and cutting

speeds (higher than recommended), or in scenarios where the part must be cooled to avoid part

8

distortion and prevent ignition of chips. The cutting fluids to be used, should generally be mineral oils

with low viscosity, preferably near 55 SUS, maximum free acid content of 0.2% and a minimum

flashing point around 135 °C [13].

The operating parameters to machine magnesium alloys, should be chosen carefully and in

light of what has been stated, cutting speeds and feeds should be as high as possible, and as safety

measures, the workstation should always be kept clean, smoking or open flames must be prohibited in

the working zone and an adequate supply of fire extinguishing agents, should be present, namely

class D, with powders G-1 or Met-L-X [13] and [14].

2.1.4 Applications of magnesium alloys

The alloy category to be used depends totally on the component to be manufactured, which in

turn depends on the application within the industry where it is to be applied. Magnesium alloys are

used in a wide range of industries, such as: Aerospace, Automotive, Medical, Electronic, Sports and

others [5], [15] and [16]. From the aforementioned branches of application, the main consumers of

magnesium alloys are the aerospace and automotive industries.

2.1.4.1 Aerospace industry

Magnesium alloys have been used in the aerospace industry for quite some time, mainly used

in military aircrafts such as the Sikorsky S-56 (Figure 2.2a), the Lockheed F-80C (Figure 2.2b), the

Convair B-36 Peacemaker (Figure 2.2c) or even the Tupolev TU-95MS (Figure 2.2d).

(a) (b)

(c) (d)

Figure 2.2 – Military aircraft applications that employ Magnesium (a) Sikorsky S-56 [17]; (b) Lockheed F-80C [18]; (c) Convair B-36 Peacemaker [19]; (d) Tupolev TU-95MS [20]

In the Sikorsky helicopter, magnesium alloys were found in the fuselage and the housing of

the main gearbox [17]. The Lockheed F-80C, was completely built with magnesium [15], [21], [22]. The

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American bomber, Convair B-36 had an impressive 8600 kg of magnesium [5], [15]. A considerable

amount of magnesium could also be found on the Tupolev TU-95MS aircraft, with about 1550 kg of

magnesium [15].

Nowadays magnesium alloys are broadly used in the aerospace industry, maintaining its

presence in the branch of defence, particularly in the manufacturing of UAV’s [23] (Unmanned Aerial

Vehicle) also known as Drones, or in other non-structural aircraft application such as cast transmission

housings for example [15].

2.1.4.2 Automotive industry

The usage of magnesium alloys in the automotive industry comprises the motorsport branch

and also motorcycle manufacturers, since these fields are also using magnesium parts presently.

The first noteworthy use of magnesium alloys in the automotive industry, dates back to 1938,

with its use in the Volkswagen Beetle, which was designed by Ferdinand Porsche, (Figure 2.3a). This

vehicle possessed more than 20 kg of magnesium made up from the transmission housing, the

crankcase and other smaller parts, all obtained through casting [5]. A few other significant applications

of magnesium alloys in the automotive industry came in the 1950’s, specifically with the Allard sports

car (Figure 2.3b), and also the 1955 Mercedes-Benz 300 SLR (Figure 2.3c) [5].

Figure 2.3 – Automotive application of magnesium alloys (a) 1938 VW Beetle [24]; (b) Allard sports car [5]; (c) Mercedes-Benz 300 SLR [5]

The main application for the latter two examples was in body components, mainly made out of

magnesium alloy sheet which contributed considerably to weight reduction, and in the case of the

Allard, the global weight of the body with doors and bonnet reached 64 kg.

Magnesium alloys are becoming more and more common in the automotive industry and there

are a few applications in which this can be observed. There is a strong presence of magnesium alloy-

based parts in the drivetrain, with components such as gearbox housings, intake manifolds,

crankcases, cylinder head covers and even engine blocks as shown in Figure 2.4.

10

Figure 2.4 – Drivetrain applications of Mg alloys (a) Mercedes 7-speed automatic transmission housing [7]; (b) Audi V8 intake manifold [5]; (c) Mercedes M291 prototype crankcase [5]; (d) BMW

cylinder head cover [5]; (e) Mercedes M291 prototype engine block [5]

Also, magnesium alloys are used for interior components, such as steering wheel armatures,

and other steering system components, seat frames, instrument panel trims and console frames. Body

parts, such as doors, tailgates, roofs or bonnets can also be manufactured in magnesium alloy. Some

of these examples are shown in Figure 2.5.

Figure 2.5 – Automotive applications (a) Steering wheel armature [5]; (b) Seat frame [5]; (c) Inner door panel [25]; (d) Tailgate [5]; (e) Bonnet inner part [5]

Magnesium alloys are found very useful in motorsport and other high performance road

vehicles, namely in Formula 1, where the wheels should be manufactured in magnesium alloys (AZ70

or AZ80) as specified in the technical rule book for the 2015 season [26], and also Italian high

performance motorcycle manufacturer MV Agusta, currently employs magnesium alloys in quite a few

drivetrain components, and has used it in swingarms in past models.

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2.1.4.3 Other applications

Magnesium alloys can be found in many other fields, particularly medical, electronic and

sports. The main reasons behind the presence of magnesium alloys in the fields previously

mentioned, is due to its characteristics, specifically its density, heat dissipation and improved

mechanical resistance when compared to plastics that are replaced or compete with magnesium.

Medical applications of magnesium alloys consist mainly of implants, due to the fact that magnesium

emulates bone behaviour consistently. In electronic devices like laptops, cellular phones and other

handheld products, magnesium is used more commonly to replace plastics, providing better

mechanical properties, while maintaining if not improving weight savings. In sports, magnesium alloys

present a strong alternative once again due to its low density, which allows for lighter equipment to be

produced, specifically bicycle frames, tennis rackets and golf clubs [16].

2.2 Fatigue

According to the American Society for Testing and Materials (ASTM) Standard E 1823, fatigue

is defined as “the process of progressive localized permanent structural change occurring in a material

subjected to conditions that produce fluctuating stresses and strains at some point or points and that

may culminate in cracks or complete fracture after a sufficient number of fluctuations”. From the

definition, it can be understood that fatigue is a phenomenon of great importance, especially due to

the fact that many systems in everyday life are subjected to fatigue, and therefore may put human

lives at stake. It is of the utmost importance, to explore, study, understand and characterize the

behaviour of materials subjected to fatigue.

2.2.1 Historical summary

The majority of the cases presented in this section were based on Walter Schütz’s paper [27].

In 1837, Albert published the first fatigue test results known, to obtain said results, he built a

test machine for conveyor chains which failed during service in the Clausthal mines.

In 1842, Rankine, well known for his contributions in thermodynamics, after studying the

fatigue strength of railway axles, proposed that these should be forged with a hub of enlarged

diameter and large radii. This year was marked by the catastrophic fatigue failure of a locomotive axle,

on 5th October in Versailles.

In 1853, Morin analysed reports related with the axles of horse-drawn mail coaches, which

stated that the mentioned axles, should be replaced after 60000 km, while axles with service lives

extended to 70000 km should be thoroughly inspected. The cracks discovered by inspection were

found mostly in section changes.

In 1854, the term “fatigue” was used for the first time by Braithwaite. In his papers, he

described fatigue failures of brewery equipment, water pumps, propeller shafts, crankshafts, railway

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axles and many other applications. He also discussed allowable stresses for components subjected to

fatigue.

In the period comprised between 1858 and 1870, Wöhler performed tests on railway axles, to

measure the service loads with self-developed deflection gauges in 1858 and 1860. The results of the

fatigue tests with railway axles were published in 1860, however Wöhler built a new machine to test

axial-bending and torsion tests on different notched and unnotched specimens. Finally in 1870,

Wöhler presented a final report with the main conclusions from his studies, in which he stated that

stress amplitude is the most important parameter for fatigue life, but a tensile mean stress also has a

harmful influence.

In 1886, Bauschinger, professor of mechanics at the currently named Technical University of

Munich, named the Bauschinger effect, which in his own words is “the change of the elastic limit by

often repeated stress cycles”.

In 1898, Kirsch calculated a stress concentration factor of 3.0 for a cylindrical hole in an infinite

plate.

In 1903, Ewing and Humfrey observed slip bands on the surface of rotating bending

specimens.

Between 1905 and 1925, there were many contributions to the topic of fatigue by British and

American engineers/scientists and the first full scale fatigue test with a large aircraft component took

place at the Royal Aircraft Establishment in the United Kingdom.

In 1910, Basquin represented the data from Wöhler’s tests in the form log (𝜎𝑎) on the ordinate,

log (𝑁) on the abscissa, describing it with the following expression:

𝜎𝑎 = 𝐶𝑁𝑛

which remains in use today. The values of 𝐶 and 𝑛 were given by Basquin, based mostly on

results obtained by Wöhler.

In 1917, professor Haigh, well known for his work in fatigue, mentioned “corrosion fatigue” for

the first time.

In 1920, Griffith of the Royal Aircraft Establishment, developed the basis of fracture mechanics

and showed through testing on brittle material glass that small crack-like scratches reduced the

breaking strength significantly and that the crack size also had an influence.

In 1924, Gough’s book “The Fatigue of Metals”, mentions the influence of surface roughness

on fatigue limit and also the stress concentration factors of V shaped notches based on the results

obtained by Coker. Also in 1924, Palmgren authored a well-known paper that contained the Palgrem-

Miner rule and a four-parameter equation extending from the tensile strength to the fatigue limit for the

SN curve.

In 1929 McAdam performed many corrosion-fatigue tests.

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In 1937 Neuber published the first comprehensive book covering the theoretical calculation of

stress concentration factors, and fatigue stress concentration factors.

From 1939 to 1945, Gassner defined the topic of operational fatigue strength

(Betriebsfestigkeit in his native German), which consists of dimensioning a component for finite, but

sufficient fatigue life under variable loads.

In 1954, Coffin and Manson defined the field of Low-Cycle Fatigue (LCF) by describing the

behaviour of metallic materials under cyclic inelastic strain amplitudes through a four-parameter

formula.

In 1955 Sines performed experimental tests with alternated biaxial loading and simple

combinations of static and alternated stresses. His conclusions stated that for brittle materials, shear

cyclic stress appeared to be the cause for fatigue failure, even though yield occurred near the

maximum theoretical normal stress, [28].

In 1956, Findley reviewed experimental tests in fatigue concluding that none of the results

obtained so far, were against the shear stress limit. Also, Findley extended some yield criteria to

fatigue analysis. Based on his work, he developed a multiaxial fatigue model which takes into account

the influence of normal stress, which in turn occurs on the maximum shear stress plane.

In 1962, Paris in his Ph.D. Thesis, stated that fatigue crack propagation could be described by

an equation, nowadays known as Paris’ Law, which relates the crack growth rate, with the stress

intensity factor.

In 1967, Miller presents a modified octahedral shear stress criterion, which takes into account

the effects of principal axis rotation in nonproportional loadings, [28].

In 1968, Elber in his Ph.D. Thesis noticed that after a high tensile load, the crack closes before

the load is reduced to zero, this phenomenon is nowadays known as “crack closure”.

Between 1969 and 1974, the American Society of Mechanical Engineers (ASME), found a

quicker and more conservative method to apply in the design of pressure vessels. In 1974, the von

Mises criterion was substituted by the Tresca criterion, regarding the design of pressure vessels, [28].

In 1970, Smith, Watson and Topper (SWT) presented a new multiaxial fatigue model,

applicable to materials that mainly fail due to crack growth on planes of maximum tensile strain or

stress, [29].

In 1973 Brown and Miller proposed a theory in which both cyclic shear strain and normal strain

on the plane of maximum shear must be considered, since according to this theory, cyclic shear strain

will help nucleate cracks, and normal strain will assist crack growth. This theory also suggests the

terms Case A and Case B cracks, depending on the nucleation and growth of the cracks, [29].

In 1974, the United States Air Force introduced a new structural specification designated

“Damage Tolerance Requirements”, which considers that crack-like defects exist in the components

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from manufacture onwards, in all critical points of the structure. To stay in accordance with this

requirement, the manufacturer must prove by test and calculation, that the component has enough

static strength (damage tolerance) and sufficient life, in the assumed cracked condition.

In 1975, Grubisic and Simbürger observed the effects of out-of-phase loading on the biaxial

fatigue strength of carbon steel, using thin-walled cylindrical specimens, and the results showed that

phase difference between shear and normal stresses can have a large influence in fatigue life, [28].

In 1976, Blass and Zamrik subjected specimens made of AISI (American Iron and Steel

Institute) 304 stainless steel to simultaneous tension-compression and alternating torsional loads at

different temperatures and with different shear strain ratios. They concluded that a fatigue failure

criterion based on shear and normal strains acting on the plane of maximum shear strain would be

more suitable than other criteria based on equivalent strain and other common measures, [30].

In 1977, Kanazawa et al. performed low-cycle fatigue experimental tests on a 1% Cr-Mo-V

steel subjected to cyclic axial and torsional loads with various phase combinations, and noticed that

endurance and direction of crack growth was dependent on the strains acting on maximum shear

planes, and also that fatigue life is reduced by out-of-phase loads, [31].

In 1979, Kanazawa et al. continued on the work presented in 1977, and analysed the cyclic

deformation under out-of-phase loads of the same 1% Cr-Mo-V steel, subjecting the specimens to

combined axial and torsional loads. The results obtained showed that the hysteresis loop for the out-

of-phase cyclic loads is quite different from the in-phase one, [32].

In 1981, Garud suggested a new approach to multiaxial fatigue analysis, based on an energy

model. Garud concluded that traction work is more damaging than shear work, and he obtained

suitable correlations for both proportional and nonproportional loading conditions for Cr-Mo-V steel.

However, this model is not apt for High-Cycle Fatigue (HCF) studies, due to the fact that the work per

cycle is quite small becoming quite difficult to compute with accuracy, [28].

In 1988, Fatemi and Socie suggested an alteration to the model proposed by Brown and

Miller, in which the normal strain would be substituted by the normal stress, the reason for this, lies in

the fact that tensile stresses will separate crack surfaces and reduce frictional forces, [33].

In 1989, Dang Van presented an endurance limit criterion based on the microstresses in a

critical volume. Fatigue crack nucleation is a local process and starts in grains that have suffered

plastic deformation and have formed slip bands. Due to this, Dang Van suggests that microscopic

shear and hydrostatic stresses are an important parameter, [29].

In 1993, Liu and Zenner, proposed a criterion that consists of a double integral. After carefully

reviewing previous work, Liu and Zenner verified that there were two methods to formulate a multiaxial

fatigue damage parameter, either by an integral formulation or by a critical plane formulation. They

concluded that integral formulation allowed to compute the damage parameter at all planes of the

critical volume, while the critical plane formulation, only takes into account the plane where the

damage parameter is maximum, [34].

15

Between 1995 and 1997, Papadopoulos presented a microscopic integration model, and a

critical plane model, respectively. The latter, is known as Minimum Circumscribed Circle (commonly

abbreviated MCC), and it allows the estimation of the shear stress amplitude, [35].

Also in 1997, Palin-Luc and Morel concluded that the HCF model was not enough to explain

all the phenomena observed when performing experimental tests. Because of this, these authors

proposed a model consisting of the analysis of a volume around a critical point which takes into

account the influence of the crack propagation initiation. The damage parameter is calculated per

cycle, and it is the energy density of elastic volumetric deformation that exceeds a limit value. This

value depends on the material and according to the authors, this value can be seen as a limit for

damage non-propagation, [34].

In 2000, Freitas, Li and Santos, suggested a new damage parameter that was based on

Papadopoulos’ MCC. This new parameter was designated Minimum Circumscribed Ellipse (MCE) and

it takes into consideration the nonproportional loading effects that were left out of the MCC model,

[36].

In 2003, Reis, Li and Freitas analysed the effect of nonproportional loading in 42CrMo4 alloy

steel, and concluded that the loading path has great influence in fatigue life, [37].

In 2004, in his Ph.D. Thesis, Reis studied the behaviour of proportional and nonproportional

loads on steel, and concluded that the MCE model provided the best results, [38].

In 2005, Wang and Yao, concluded that for a case of multiaxial load, with the same equivalent

von Mises stress, fatigue life was shorter with the increase of nonproportionality between loads,

finding the minimum fatigue life with 90° out of phase loads. The conclusions presented were based

on experimental tests performed on LY12CZ Aluminium alloy specimens. From this study, the authors

proposed a new critical plane damage parameter based on shear stress range and normal stress

range which acts perpendicularly to the critical plane, [39].

In 2006, Hasegawa et al. presented the results of their work on stress controlled, uniaxial Low-

Cycle Fatigue (LCF) tests, performed on extruded AZ31 Magnesium alloy, which allowed them to

conclude that compression yielding is easy due to twinning which leads to asymmetric hysteresis

curves; and the specimens also tend to deform quasi-elastically during unloading from compression,

which makes the plastic strain amplitude smaller to the maximum one in the hysteresis curve, [40].

In 2008, Tsushida et al. studied the relation between grain size and fatigue strength, for the

AZ31 Magnesium alloy, concluding that twinning under the fatigue test depends on the grain size, and

it affects the fatigue life of the alloy, [41].

Also in 2008, Begum et al. noticed the asymmetrical cyclic behaviour when performing strain

controlled axial tests on an AZ31 Magnesium alloy, [42].

In 2009, Tokaji et al. studied fatigue crack propagation and fracture mechanics for wrought

AZ31 and AZ61 magnesium alloys, in different environments, namely laboratory air, dry air and

16

distilled water. The fractography analysis allowed them to conclude that the fracture mechanisms that

operated in laboratory air and in distilled water were different, possibly due to hydrogen embrittlement

and anodic dissolution, respectively, [43].

Also in 2009, Reis et al. presented the results of their study on crack initiation and growth path

under multiaxial fatigue loading in three structural steels: Ck45, 42CrMo4 and AISI 303. They verified

that the different materials have different crack orientation under the same loading path, due to

different plasticity behaviour and sensitivity to nonproportional loads between the materials, [44].

In 2010, Bernard et al. conducted experimental tests to evaluate the fatigue properties of an

extruded Mg-3Al-0Mn magnesium alloy component. From the work performed they were able to notice

that specimens with smaller grain sizes had greater fatigue life, while in contrast, the larger the

average grain size, the lower the fatigue life, [45].

Also in 2010, Albinmousa et al. investigated the multiaxial fatigue behaviour of extruded

AZ31B magnesium alloy, with cyclic tension-compression, cyclic torsion, proportional and

nonproportional experimental tests. They observed that for cyclic axial tests the alloy shows

asymmetrical cyclic behaviour due to twinning, while the behaviour found on cyclic torsional tests was

symmetric. They proposed an energy-based model to correlate the tension, shear and multiaxial

results, [46].

Still in 2010, Reis et al. presented their work on the investigation of mean stress effects during

cyclic stress/straining of 42CrMo4 steel. Both numerical methods and experimental tests were carried

out in this work. The numerical results agreed with the experimental results, [47].

In 2011, Albinmousa et al. published more work directed at the behaviour of AZ31B

magnesium alloy. One of the studies presented by the authors, investigated the multiaxial cyclic

behaviour of extruded AZ31B magnesium alloy using tubular specimens machined from large

extruded sections, subjected to two loading conditions: axial and torsional. They concluded that

twinning has a large influence in deformation under multiaxial loading and nonproportionality has no

significant influence on fatigue life, [48].

The other work presented by Albinmousa et al. in the same year, consisted of pure cyclic axial

and pure cyclic torsional behaviour characterization, through testing of tubular specimens machined

from extruded AZ31B magnesium alloy. The authors concluded that the material in question,

experiences significant cyclic hardening and plastic strain reduction when subjected to cyclic axial

loading while the cyclic shear hardening is less pronounced, [49].

Still in the year 2011, Zeng et al. published their investigation on the influence of frequencies

on fatigue crack propagation rates of two magnesium alloys: AZ80 and AZ61. The results obtained

allowed them to conclude that the fatigue crack propagation rates on both alloys would increase with a

reduction of the frequency and that the cyclic loading frequency has a significant impact on the strain

rate which leads to a change in the mechanical properties of the specimens, [50].

17

In 2012, Anes et al. presented the study conducted to evaluate the mechanical behaviour of

AZ31 magnesium alloy subjected to low-cycle fatigue. AZ31B cylindrical specimens were subjected to

a cyclic uniaxial load and several total strain amplitudes. The authors observed material softening at

tension and hardening at compression, for lower total strain amplitudes, [51].

In 2013, Anes et al. published the results of their investigation on crack path evaluation for two

different microstructures: BCC and HC, performing tests on two different materials, specifically

42CrMo4 with a BCC microstructure, and magnesium alloy AZ31B-F with a HC microstructure. They

observed that for multiaxial loading conditions the loading path trajectory had a significant influence on

stress concentration factors, [52].

Also in 2013, Anes et al. presented a new approach to determine stress scale factors (SSF)

for multiaxial fatigue loadings on any materials, with an algorithm based on S-N results from specific

loading paths, [53].

Still in 2013, Itoh et al. proposed a method to determine the principal stress and strain ranges

along with mean stress and strain under proportional and nonproportional loading in 3D stress and

strain space, [54].

Another investigation presented in 2013 was Shamsaei and Fatemi’s study on small fatigue

crack growth under multiaxial stresses. The authors carried out experimental tests on 1045 and 1050

steels, 304L stainless steel and Inconel 718. The authors observed that a compressive normal stress

on the maximum shear plane contributes to the deceleration of crack growth, while a tensile normal

stress accelerates crack growth and also that crack surface roughness resulted in friction-induced

closure, [55].

In 2014, Cláudio et al. presented the results of a study regarding in plane biaxial fatigue of

cruciform aluminium specimens. The authors concluded that most of the criteria used, yielded non-

conservative results with the exception of the MCE (Minimum Circumscribed Ellipse) method, which

provided better results, [56].

Also in 2014, Baptista et al. presented the results of the study performed to optimize the

design of cruciform specimens for in-plane biaxial fatigue testing, [57].

Still in the year of 2014, Anes et al. proposed a new approach to evaluate non-proportionality

in multiaxial loading. The authors carried out tests on three different steels, specifically Ck45,

42CrMo4 and AISI 303. The results allowed to conclude that a constant damage scale factor between

axial and shear stress is not suitable to quantify different damage mechanisms in proportional and

non-proportional loading paths. The proposed factor, Y factor, allowed achieving good results in

fatigue life correlations, [58].

Another work of interest in 2014, presented by Anes et al. demonstrated the application of

previously developed models (MCE and SSF) to experimental data obtained by other research

groups. The results obtained for the aforementioned models were very acceptable, [59].

18

Anes et al. proposed a new cycle counting method and a fatigue life evaluation criterion in

2014. The proposed models were compared with other well-known models and were correlated with

fatigue data, yielding acceptable results, [60].

In 2015, Anes et al. presented their investigation on damage accumulation under variable

amplitude loading conditions, employing Palmgren-Miner’s rule, Morrow’s rule and the SSF criterion

as a damage parameter to compare the results, [61].

Still in 2015, Li et al. proposed a new fatigue life prediction model composed of three parts:

multiaxial fatigue life surface, a new path-dependent factor for multiaxial high cycle fatigue and a

material parameter that takes into account the material sensitivity to non-proportional loading, [62].

2.2.2 Multiaxial Fatigue

Generally, most of the engineering components found in every field of application are

subjected to fatigue loadings, and in most cases the loadings are multiaxial. The multiaxial states of

stress that arise from the loading on a structure/component, present a much more difficult assessment

of fatigue life for said structure/component.

When a component is subjected to cyclic stresses, usually both the orientation of the principal

axes and the magnitude of the stresses change with time, and due to this, the study of multiaxial

fatigue becomes more difficult and less predictable. Multiaxial loadings can be classified as

proportional, and nonproportional, depending on a combination of factors.

2.2.2.1 Proportional Loading

In a proportional loading case the cyclic stresses are applied in-phase, and may or may not

have the same amplitude. In Figure 2.6, [29], the concept of proportional loading is illustrated

considering a shaft subjected to in-phase axial and shear cyclic stresses. If a new coordinate system,

X’-Y’, is defined so that 𝜎𝑋′ = 𝜎1 , and it is kept fixed relatively to the shaft’s axes X-Y, one can

observe that the X’ axis always coincides with the principal normal stress axis. Quoting Socie and

Marquis, [29], “proportional loading is defined as any state of time varying stress where the orientation

of the principal stress axes remained fixed with respect to the axes of the component.”

19

Figure 2.6 – Proportional multiaxial loading, [29].

2.2.2.2 Nonproportional loading

In the case of nonproportional loadings, the cyclic stresses are applied out-of-phase, or as in

Figure 2.7, [29], one of the stresses (axial in this case) is kept constant, while the applied shear stress

is cyclic. Considering once again a X’ axis, fixed relatively to X, so that 𝜎𝑋′ = 𝜎1 at point A.

Figure 2.7 – Nonproportional loading, [29].

It is possible to observe that the orientation of X’ does not coincide at all times with the

principal normal stress axis, therefore this is a nonproportional loading. Once again quoting Socie and

Marquis, [29], it is a “state of time varying stress in which the orientation of the principal stress axes

changes with respect to the axis of the component.”

20

2.2.3 Material Behaviour

Understanding the behaviour of the material subjected to fatigue is of high importance to make

a correct evaluation of fatigue life for a certain component. A very important aspect which holds high

influence in fatigue damage is plastic deformation as known since the work of Bauschinger, [29].

Cyclic plastic shear deformation is the main cause for formation of slip bands, along grains

with favourable crystallographic slip planes and directions. However, due to the fact that grains have

different orientations, slip bands form only when the applied stress provokes a shear stress high

enough to begin plastic deformation (critical resolved shear stress) on a particular grain. When the

material is subjected to cyclic loads, the referred slip bands tend to grow and coalesce forming a

significant fatigue crack.

The crack nucleation process is highly influenced by grain boundaries, precipitates, impurities

and even inclusions, and it can be explained in a simple way by the intrusion-extrusion model shown

in Figure 2.8, [29].

Figure 2.8 – Intrusion-Extrusion model that leads to slip band formation, [29].

The intrusion-extrusion model pictured above depicts the formation of slip bands as a result of

dislocation movement within individual grains, which can only happen along the favourable slip plane

of the grain, as previously stated. The dislocations move due to the cyclic shear stresses and when

the critical resolved shear stress is reached, plastic deformation takes place resulting in a permanent

offset in adjacent atomic planes. The slip bands formed by this plastic deformation can be extrusions if

they come out of the surface of the material, or intrusions in case they go into the surface of the

material, and the repeated straining of these slip bands will cause formation of a crack.

The cyclic deformation behaviour of the material can be characterized by the following

phenomena, [29]:

Isotropic hardening, to account for changes in the strength of the material;

Kinematic hardening, to model the Bauschinger effect and material memory;

21

Cyclic creep or ratcheting, to consider the increase in mean plastic strain in each cycle

during stress-controlled deformation with a mean stress;

Mean stress relaxation, to account for the relaxation of the mean stress during strain-

controlled deformation with a mean strain;

Nonproportional cyclic hardening, to model the stress increase that occurs in the

plastic regime with nonproportional loading.

The aforementioned phenomena, is analysed in more detail in the following subsections.

2.2.3.1 Isotropic hardening

Isotropic hardening characterizes the increase in yield strength of the material, due to plastic

strain. The left side of Figure 2.9 is a good illustration of this phenomenon. The stress-strain response

for isotropic hardening shows that plastic deformation of the material will start at point A, and this will

cause work hardening in the materials as the dislocations interact with each other. Considering that

upon reaching point B the load is removed, and then applied again, the material will now yield at a

new stress, 𝜎𝐵, and plastic deformation will continue on its original path. This is due to a material

characteristic designated material memory.

Figure 2.9 – Isotropic Hardening, [29].

If the material continues being loaded up to point C, then due to isotropic hardening, the new

yield strength of the material will be 𝜎𝐶. In the event of the material being loaded in compression,

yielding would occur at point D at a tension −𝜎𝐶. The von Mises yield surface is represented on the

right side of Figure 2.9, in a tension-torsion stress space. Any combination of normal and shear stress

that coincides with the yield surface will provoke plastic deformation on the material. Comparing a

situation of loading the material to point A and another in which the material would be loaded up to

point C, it can be observed that the yield surface has kept its shape, but expanded evenly in all

directions while plastic deformation occurred, [29].

22

2.2.3.2 Kinematic Hardening

The kinematic hardening phenomenon models the Bauschinger effect and the previously

stated concept of material memory. In this model the yield surface is allowed to translate in stress

space, however, it does not change in shape or size. Considering the stress-strain response on the

left of Figure 2.10 the material is loaded past its yielding point A, up to the plastic region and a stress

𝜎𝐵.

Figure 2.10 – Kinematic hardening, [29].

Contrary to isotropic hardening, the yield surface translates due to plastic deformation. If the

material is unloaded and reloaded again, the yield surface will not translate any further because this

only happens during plastic straining of the material. If the material is reloaded in the same direction, it

will now yield at 𝜎𝐵, similarly to what happened in the isotropic hardening case. Although, if the

material is loaded in compression, the response will be quite different, because reverse yielding will

happen at point C and a stress 𝜎𝐶, which corresponds to 𝜎𝐵 − 2𝜎𝑦. The translation of the yield surface

is represented by a vector, 𝛼 (as illustrated in Figure 2.10), and it occurs in the direction of plastic

strain. This is what was enumerated as Bauschinger effect, and is commonly called “backstress”.

Real materials display aspects of isotropic and kinematic hardening until they become

cyclically stable, after which they only display kinematic hardening. In most cases of interest in fatigue

analysis, a cyclically stable material is assumed, and only kinematic hardening models are used, [29].

2.2.3.3 Cyclic Creep or Ratcheting

Cyclic creep or ratcheting can be described as the accumulation of plastic deformation when a

material is subjected to a mean stress, [29]. Considering, as an example, a thin-walled tube subjected

to a static axial stress and a cyclic shear strain, which is enough to produce plastic deformation during

23

each cycle. Figure 2.11 shows a plot of shear strain vs axial strain, and it should be noted that the total

axial deformation continues to increase during each cycle.

Figure 2.11 – Ratcheting, [29].

When the loads are applied, axial and shear strains increase up to point A. The axial strain is

kept constant during the elastic portion of loading to B, after point B is reached, plastic deformation

takes place and a coupling between axial and shear strains occurs, which means that during plastic

loading, the shear loads will produce axial deformation. For the case presented, the ratcheting rate

decreases and deformation becomes stable. If large plastic strains are applied, then the ratcheting

rate increases with each cycle, [29].

2.2.3.4 Mean stress relaxation

Cyclic creep and mean stress relaxation are related. Mean stress relaxation takes place during

strain-controlled deformation with an initial mean stress which tends to zero during each loading cycle,

as shown in Figure 2.12. The rate at which the mean stress relaxation happens, is dependent on the

plastic strain range, and the initial mean stress.

Cyclic creep and mean stress relaxation models often need the input of several material

parameters, and even though many models have been developed throughout the years, none has

been universally accepted.

Figure 2.12 – Mean stress relaxation, [29].

24

2.2.3.5 Nonproportional cyclic hardening

Nonproportional cyclic hardening happens with nonproportional loadings, in which the

orientation of the main stress axes changes with time, as stated in section 2.2.2.2. Considering

alternating cycles of tension and torsion loading, and plotting the effective stress vs effective strain, for

proportional and nonproportional loadings allows to observe the effect of nonproportional hardening,

as shown in Figure 2.13, [29].

Figure 2.13 – Cyclic stress-strain curve for proportional and nonproportional loading, [29].

The impact of nonproportional cyclic hardening is highly dependent on the material’s

microstructure, and the slipping mechanism developed by the material. Loading history also has

influence on nonproportional hardening, and nonproportionality factors are used to interpolate

between in-phase (proportional) and 90° out-of-phase (nonproportional) stress-strain curves to

achieve a stable stress-strain curve for any nonproportional loading, [29].

2.2.4 Fatigue Crack Growth

Fracture is always associated with crack growth and due to that, knowledge of the crack

growth phenomenon is of high importance.

Considering fracture mechanics terminology, the type of load applied to a crack can be

classified as a mode. Figure 2.14 illustrates the three modes of crack loading according to fracture

mechanics.

25

Figure 2.14 – Different modes of crack loading, [63].

In Mode I the crack surface is opened due to tension loads (opening mode), and Modes II and

III are due to shear loads with Mode II being characterized by in-plane shear (sliding mode), and Mode

III by out-of-plane shear (tearing mode).

As previously discussed (section 2.2.3), cracks tend to nucleate due to cyclic shear strains.

However, once the crack reaches a large enough size, the crack growth will continue perpendicular to

the applied tensile stress, due to the fact that the crack can generate its own plasticity and carry on

growing through the grains. This concept is illustrated in Figure 2.15, where stage I refers to the initial

formation of the crack through shear, and stage II represents the crack growth perpendicular to the

applied loading.

Figure 2.15 – Stage I and II crack growth, [29].

According to Linear Elastic Fracture Mechanics (LEFM), the rate at which cracks propagate

(usually considered in mm/cycle) has a relation with the stress intensity factor range, as shown in the

log-log graph of Figure 2.16.

26

Figure 2.16 – Relation between 𝑑𝑎/𝑑𝑁 and ∆𝐾, adapted from [64].

The graph in Figure 2.16 is divided in three regions that represent different stages of crack

growth. In region I the crack propagation rate is highly dependent on the stress intensity factor (SIF)

range, and propagation may not occur or be negligible, in case the stress intensity factor range

threshold is not reached, this threshold value is represented by ∆𝐾𝑡ℎ. This region shows non-

continuum behaviour and is highly influenced by the microstructure of the material, mean stress and

the environment. Region II shows a linear relation between 𝑑𝑎/𝑑𝑁 and ∆𝐾, which can be written as a

power law known as Paris’ Law (2.1):

𝑑𝑎

𝑑𝑁= 𝐶∆𝐾𝑚 (2.1)

Where 𝑑𝑎/𝑑𝑁 is the crack propagation rate, ∆𝐾 is the stress intensity factor range, and 𝐶 and

𝑚 are material constants. In this region, continuum behaviour is verified and certain combinations of

mean stress, frequency and even environment tend to have a large influence.

Region III shows an acceleration of crack growth rate with ∆𝐾 and unstable crack propagation,

which leads to catastrophic failure. The transition to region III is related to the increase in SIF range

which approaches a critical value of stress intensity factor, usually represented as 𝐾𝐼𝑐 or 𝐾𝑐 and

denominated fracture toughness which is a material property. As opposed to the other two regions,

region III has little influence of the environment, but is still largely influenced by the microstructure of

the material and mean stress as well as the remaining thickness of the component or specimen.

The stress intensity factor range can be computed in the following way:

∆𝐾 = 𝑌∆𝜎√𝜋𝑎 (2.2)

Where 𝑌 is a shape factor, ∆𝜎 is the applied stress range, and 𝑎 is the crack size.

27

2.2.5 Fatigue Life

The concept of fatigue life is of high importance for the design of mechanical structures and

components subjected to fatigue loadings. Based on the previous discussion regarding crack

nucleation and growth, fatigue life can be schematically illustrated by Figure 2.17.

Figure 2.17 – Schematic representation of fatigue life, [65].

It is worth to mention that the number of cycles, 𝑁, in region III (Figure 2.17) is considered

negligible due to the unstable behaviour that leads to catastrophic failure, shown in this region of crack

propagation. Since the added contribution of the cycles spent during initiation, region I and region II is

much larger than the portion spent in region III, it is imperative to attempt to maximize it through

adequate design philosophies when designing a component subjected to fatigue loading.

2.2.6 Design Theories

The design theories employed in designing of mechanical components, structures or systems

have evolved during the years, although some approaches might seem obsolete, there are

applications where a much more conservative approach is still used.

2.2.6.1 Infinite-Life Design

This is the oldest design theory and it considers that all stresses or strains are elastic and

sufficiently below the endurance limit, in order to ensure “infinite” service life for the component or

structure, [1].

2.2.6.2 Safe-Life Design

In this theory, the design of a component establishes a period of finite life, called “safe-life” for

which the probability of crack initiation and growth is remote, due to considering that overloads might

occur only occasionally during the life of the system, [1].

2.2.6.3 Fail-Safe Design

Essentially, this theory requires that a structure or system does not fail if a single component

fails. It takes into account that the structure will possess enough residual strength after partial or total

failure of a single component, [1].

28

2.2.6.4 Damage-Tolerant Design

This theory is an improvement on Fail-Safe Design. It considers that cracks will exist due to

either fatigue or processing, and cracks are allowed to grow, while being periodically inspected and

monitored according to the foreseen propagation tendency. Figure 2.18 schematically shows the

concept of Damage-Tolerant Design.

Figure 2.18 – Schematic representation of the Damage-Tolerant Design concept, [66].

During the time period indicated as “duration” (Figure 2.18) several periodic inspections, (by

means of Non-Destructive Testing), must be carried out with shorter and shorter intervals between

them, to ensure that crack propagation is within the foreseen values. This theory intends to determine

the effect of cracks in the residual strength of the component and also determine crack growth as a

function of time, [1].

2.2.7 Fatigue Models

An important aspect of material behaviour under multiaxial fatigue is proper fatigue life

prediction through multiaxial fatigue models and damage parameters. This subsection presents an

overview of the fatigue models considered in this study.

2.2.7.1 Findley

The Findley model is a critical plane model which considers that the normal stress on a shear

plane has a linear influence on the alternating shear stress, as given by (2.3).

(𝛥𝜏

2+ 𝑘𝜎𝑛)

𝑚𝑎𝑥= 𝑓 (2.3)

Where 𝑓 represents the model’s damage parameter, 𝛥𝜏 is the shear stress range (hence

dividing by 2 to obtain the shear stress amplitude), 𝜎𝑛 is the normal stress and 𝑘 is a constant that

should be determined through experimental tests consisting of two or more stress states.

This model allows the identification of critical planes, which are the planes within the material

that are subjected to a maximum value of a damage parameter (2.3). The damage parameter of

Findley’s model is dependent on the combined action of the alternating shear stress and the maximum

normal stress, and this combined action is responsible for fatigue damage.

29

Since this model identifies the critical planes in the material as planes oriented at an angle 𝜃,

then (2.3) can be written in the form:

max𝜃(𝜏𝑎 + 𝑘𝜎𝑛,𝑚𝑎𝑥) (2.4)

As previously mentioned, to determine the constant 𝑘, experimental tests must be carried out

with two or more stress states such as pure torsion (2.5) and pure axial/bending loading (2.6).

√1 + 𝑘2 𝛥𝜏

2= 𝑓 (2.5)

√((𝜎𝑎)2 + 𝑘2(𝜎𝑚𝑎𝑥)2 + 𝑘𝜎𝑚𝑎𝑥 = 2𝑓 (2.6)

Combining equations (2.5) and (2.6) yields equation (2.7), which can be used to obtain the

material constant, 𝑘.

𝜎𝑎,𝑅=−1

𝜏𝑎,𝑅=−1

=2

1 +𝑘

√1 + 𝑘2

(2.7)

2.2.7.2 Brown and Miller

Brown and Miller’s model suggests that cyclic shear and normal strains on the plane of

maximum shear should be considered, based on the principle that cyclic shear strains will help crack

nucleation while normal strain contributes to crack growth. Considering crack nucleation and growth,

Brown and Miller proposed two types of cracks: Case A and Case B cracks, shown in Figure 2.19.

Figure 2.19 – (a) Case A; (b) Case B cracks, [29].

Cracks such as the ones depicted in Figure 2.19a (Case A cracks), tend to be shallow and

possess a small aspect ratio, as a consequence of shear stresses acting on the free surface, parallel

to the length of the crack. This type of crack is always related with combined tension-torsion loadings.

Case B cracks (Figure 2.19b) can be described by the intrusion-extrusion model (section 2.2.3) and

30

these cracks tend to grow into depth, due to the acting shear stresses in biaxial tension. Generally

these cracks intersect the surface at an angle of 45°.

Brown and Miller proposed different criteria for each of the crack types mentioned previously,

which are given in equations (2.8) and (2.9) for Case A and B respectively.

(∆𝛾

𝑔)

𝑗

+ (𝜀𝑛

ℎ)

𝑗

= 1 (2.8)

∆𝛾

2= 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡

(2.9)

Where 𝑗, 𝑔 and ℎ are constants determined experimentally, with 𝑗 ranging from 1 for brittle

materials, to 2 for ductile materials. At a later point in time, Kandil, Brown and Miller suggested a

modified formulation for Case A cracks, which is given in (2.10).

∆𝛾 = (∆𝛾𝑚𝑎𝑥𝛼 + 𝑆∆휀𝑛

𝛼)1

𝛼 (2.10)

Where ∆𝛾 is the equivalent shear strain range, 𝑆 is a material parameter that weighs the

contribution of normal strain on crack growth and can be obtained by correlating axial and torsion

data, ∆𝛾𝑚𝑎𝑥 is the maximum shear strain range and ∆휀𝑛

is the normal strain range on the plane

subjected to the maximum shear strain range. Considering the equivalent shear strain amplitude given

by (2.11),

∆�̂�

2=

∆𝛾𝑚𝑎𝑥

2+ 𝑆∆휀𝑛

(2.11)

Then the critical plane can be found through equation (2.12).

max𝜃 (𝛥𝛾𝑚𝑎𝑥

2+ 𝑆𝛥휀𝑛) (2.12)

2.2.7.3 Fatemi and Socie

The model developed by Fatemi and Socie (FS), is built on the same grounds as the Brown

and Miller model, however, this model proposes that the normal strain terms should be replaced by

the normal stress. Figure 2.20 provides a good illustration of the damage model proposed by Fatemi

and Socie.

Shear loading will cause friction forces due to the fact that crack surfaces have irregular

shapes, and this will reduce crack tip stresses, obstructing crack growth and increasing fatigue life

consequently. When tensile stresses and strains are present, the crack surfaces will be separated and

the friction forces caused by shear loadings will be reduced contributing to crack growth.

31

Figure 2.20 – Fatemi and Socie’s model schematic illustration, [29].

The critical plane according to this model can be obtained by equation (2.13).

max𝜃 [∆𝛾

2(1 + 𝑘

𝜎𝑛,𝑚𝑎𝑥

𝜎𝑦)] (2.13)

Where ∆𝛾 is the shear strain range, 𝜎𝑦 is the material’s tensile yield stress, 𝜎𝑛,𝑚𝑎𝑥 is the

maximum normal stress on the plane of maximum shear strain and 𝑘 is a material constant.

2.2.7.4 Smith, Watson and Topper

The Smith, Watson and Topper (SWT) model is an alternative damage parameter for

materials that fail predominantly by crack growth on planes with maximum tensile stress or strain, as

opposed to previously discussed models for which the main failure mechanism is shear (nucleation

and growth).

This model is presented as adequate for materials for which cracks nucleate by shear, but

early life is heavily influenced by crack growth on planes perpendicular to the maximum principal

stress and strain (Mode I). Figure 2.21 shows the mechanism considered by this model.

Figure 2.21 - Smith, Watson and Topper crack growth mechanism, [29].

32

The SWT model is a critical plane model based on principal strain range and maximum stress

on the corresponding plane, and can be found through (2.14).

max𝜃 (𝜎𝑛,𝑚𝑎𝑥∆𝜀1

2) (2.14)

Where 𝜎𝑛,𝑚𝑎𝑥 is the maximum stress and ∆휀1 is the principal strain range.

2.2.7.5 Liu I and Liu II

Liu’s model is based on virtual strain energy (VSE) and is composed by an elastic part and a

plastic part and can be considered a critical plane model due to the fact that the work quantities are

defined for particular planes in the material.

The aforementioned virtual strain energy is given by the sum of elastic and plastic work

components, which are represented in Figure 2.22.

Figure 2.22 - Elastic and plastic strain energies, [67].

For the case of multiaxial loading, this model considers two modes of failure: Mode I, caused

by tension loading and Mode II caused by shear loading, and VSE can be computed with (2.15).

∆𝑊 = ∆𝑊𝐼 + ∆𝑊𝐼𝐼 (2.15)

Where ∆𝑊 is the virtual strain energy of a specific plane within the material, ∆𝑊𝐼 is work

quantity related to Mode I and ∆𝑊𝐼𝐼 is the work quantity related to Mode II. The material plane with the

maximum virtual strain energy is where failure is expected to occur.

The axial work quantity, ∆𝑊𝐼, is determined by finding the plane with the maximum axial work

and adding the respective shear work, as shown in (2.16).

∆𝑊𝐼 = (∆𝜎𝑛∆휀𝑛)max + (∆𝜏∆𝛾) (2.16)

Analogously, ∆𝑊𝐼𝐼 is computed by identifying the plane with the maximum shear work and

adding the respective axial work, as given in (2.17).

∆𝑊𝐼𝐼 = (∆𝜎𝑛∆휀𝑛) + (∆𝜏∆𝛾)max (2.17)

33

2.2.7.6 Chu, Conle and Bonnen

The Chu, Conle and Bonnen parameter is similar to the model presented by Liu in the way

that it combines normal and shear work, however, this model uses maximum stresses instead of

stress ranges, and strain amplitudes instead of strain ranges.

The maximum work can be computed through equation (2.18).

∆𝑊∗ = (𝜏𝑛,max ∆𝛾

2+ 𝜎𝑛,max

∆𝜀

2)

max (2.18)

34

3 Material, Equipment and Methods

This chapter presents the properties and chemical composition of the material used in this

study and also the equipment used for the experimental tests as well as brief description of the test

performed.

3.1 Material

The magnesium alloy presented and studied in this thesis, is designated as AZ31B-H24,

which according to the explanation previously stated (section 2.1.2), corresponds to an alloy whose

main alloying elements are Aluminium and Zinc, with around 3 and 1% respectively, and a treatment

identified as H24. The properties of interest for the magnesium alloy AZ31B-H24 are summarized in

Table 3.1, [68]. The letter B indicates it is an alloy that differs slightly in composition. The range values

of the elements of this alloy are shown in Table 3.2, [68].

Table 3.1 – AZ31B-H24 properties [68]

Property Value

Density, [g cm-3

] 1.77

Hardness, Brinell 73

Ultimate Tensile Strength, [MPa] 290

Yield Tensile Strength, [MPa] 220

Elongation at break 15 %

Modulus of Elasticity, [GPa] 45

Compressive Yield Strength, [MPa] 180

Ultimate Bearing Strength, [MPa] 495

Bearing Yield Strength, [MPa] 325

Poisson Ratio 0.35

Shear Modulus, [GPa] 17

Shear Strength, [MPa] 160

Electrical Resistivity, [µΩ cm] 9.20

Heat of Fusion, [J g-1

] 340

Coefficient of Thermal Expansion (0-100 °C), [K-1

] 26.0 x 10-6

Specific Heat Capacity, [J (g °C)-1

] 1.00

Thermal Conductivity, [W (m K)-1

] 96.0

Melting Point, [°C] 605 - 630

Table 3.2 – Percentage range of the alloying elements in the AZ31B-H24 alloy [68]

Element Al Ca Cu Fe Mg Mn Ni Si Zn

Wt % Min 2.5 - - - - - - - 0.60

Max 3.5 0.04 0.05 0.005 97 0.20 0.005 0.10 1.40

35

3.2 Specimen Geometry

The specimens used to conduct this study were cruciform-shaped, as this general geometry is

adequate for in-plane biaxial fatigue testing, [29]. The specific geometry of the specimens used has

been optimized in a previous study conducted by Baptista et al. [57], and this optimization study

allowed obtaining a set of design variables which are all Pareto fronts, i.e., mathematically they are all

equal and optimum for the imposed constraints and objective functions. The main goals for the

optimization procedure were to obtain the maximum stress level at the centre of the specimen and

also guarantee a certain uniformity of that maximum stress, conditions that are favourable for crack

initiation. Two variables that define the geometry of the specimen were kept constant, namely the

specimen arm length with a value 200 mm and the arm width with a value of 30 mm, and the rest of

the geometry is defined as shown in Figure 3.1. Keeping the arm length at 200 mm was essential in

order to agree with the minimum arm length allowed by the testing apparatus; it also ensured that

more specimens could be obtained from the raw sheet.

Figure 3.1 – General geometry of the cruciform specimen

In order to comply with the main goals set for the optimization, the specimen geometry

employs a thickness reduction at the centre and elliptical fillets between the arms, as can be seen in

Figure 3.1.

The thickness reduction at the centre is generated by a revolving spline, and the whole region

is defined by three different variables: the radius of the thickness reduction region (rr), the thickness at

the centre (tt) and the spline exit angle (θ). The aforementioned variables are obtained from the

Direction 2

Transversal

Direction 1

Longitudinal

(rolling direction)

36

optimization procedure and for the particular case of the specimens used in the present work, the

spline exit angle (θ) was kept at 90° which provides a better configuration of the geometry for crack

propagation studies.

Although the centre thickness is reduced and compressive loads can be applied in testing,

buckling phenomena should only take place in the event of a crack size large enough to significantly

decrease the stiffness of the specimen.

The elliptical fillets are defined by three variables: the major radius (RM), the minor radius

(Rm) and the centre of the ellipse (dd). These parameters were also obtained by the optimization

procedure. The values of the variables identified in Figure 3.1 are summarized in Table 3.3.

Table 3.3 – Values of the design variables considered

t, [mm] RM, [mm] Rm, [mm] dd, [mm] rr, [mm] θ, [°] tt, [mm]

3,25 63,3 20,1 51,3 7,5 90 0,364

The manufacture of the specimens posed a challenge due to two fundamental conditions: the

need to minimize the waste of material per sheet, and the particularities related to machining

magnesium alloys, i. e. fire hazard as mentioned in section 2.1.3.

The first stage of the manufacture, which consisted of obtaining a general shape of the

specimen, was done by abrasive waterjet, due to the fact that abrasive waterjet differs slightly from

pure waterjet cutting. In the case of pure waterjet cutting, the water stream erodes the material, while

in abrasive waterjet cutting the water stream accelerates the abrasive particles which are responsible

for eroding the material. The width of cut (also known as kerf) for abrasive waterjet is usually small,

which allowed for a reduction in material waste, and besides the narrow kerf, abrasive waterjet also

provides a few other advantages such as non-existing heat-affected zones, no mechanical stresses

and leaves little or no burr.

The second stage of the manufacture was done in a Computerized Numeric Control (CNC)

milling machine with end mills according to the operations that were performed. The specimen shape

after the first and second stages of manufacture is shown in Figure 3.2.

The cruciform specimens were manufactured in a Cincinnati Arrow VMC-750 CNC milling

machine. The specimens were fixed with a previously developed mould which allows fixing by means

of vacuum granted by the existence of small channels in the mould. The vacuum pump kept the

specimens in place at a pressure of 800 mbar and over. The operations that took place in the milling

machine are listed and described in Table 3.4

It is worthy to mention that all cutting operations were performed without cutting fluid, as

recommended in [13].

37

Figure 3.2 – Specimen geometry: (a) after the first stage; (b) after the second stage.

.

Table 3.4 – Operation sequence, type of tool and cutting parameters

Operation No.

Type of tool Operation description Cutting parameters

Spindle speed [rpm]

Feed rate [mm/min]

1 ∅6 Top end mill Centre thickness reduction in spiral starting at the centre.

7000 1400

2 ∅5 Ball end mill Centre thickness reduction in spiral starting at the centre.

7000 1400

3 ∅10 Top end mill Drilling of holes at the arms’ ends.

1400 1000

4 - Cleaning of the mould, specimen flip-over and burr removal.

- -

5 ∅6 Top end mill Centre thickness reduction in spiral starting at the centre.

7000 1400

6 ∅5 Ball end mill Centre thickness reduction in spiral starting at the centre.

7000 1400

7 ∅6 Top end mill Top milling of elliptic fillets between the specimen’s arms

7000 1400

8 ∅6 Single blade top

end mill

Top milling of elliptic fillets between the specimen’s arms

7000 1400

9 - Cleaning of the mould, burr removal

- -

38

After machining and before final polishing, the specimen centre thickness was measured with

a digital indicator (with its zero point set at the highest point of the sphere) by laying the specimen on a

sphere and finding the lowest value. Due to the fact that the centre thickness of the specimen required

high precision, the acceptance of a specimen to testing, was defined as ± 0.02 mm. Table 3.5

compares the measured thickness values with the one required and displays the status of the

specimen as “Fit” or “Unfit” for testing.

After the specimen manufacturing process was complete, the polishing stage took place, using

grinding papers, going from P600 to P2500 and followed by final polishing with a 3 µm diamond

suspension.

The arms of the specimens were marked with a scribe in order to keep important data and

recognize the specimen. With that being said, the arms were marked with numbers from 1 to 4

counter-clockwise, with numbers 1 and 3 corresponding to the rolling direction, and 2 and 4 to the

transverse direction (refer back to Figure 3.1). The individual identification of the specimen was

marked in arm number 1, and was defined as “BTM2022-00X” where “X” is a number. Arm 1 also

included the measure of centre thickness for the specimen as well as the average thickness of the

arms. Arm number 3 was also marked with the measures of the first and second quadrants, measured

with a digital calliper, established by the arm numbers (between 1 and 2 is the first quadrant, and

between 2 and 3 is the second quadrant).

Table 3.5 – Measured values of centre thickness.

Specimen ID Required thickness Measured thickness Difference Status

BTM2022-002 0.364 0.326 -0.038 Unfit

BTM2022-003 0.364 0.358 -0.006 Fit

BTM2022-004 0.364 0.376 +0.012 Fit

BTM2022-005 0.364 0.370 +0.006 Fit

BTM2022-006 0.364 0.242 -0.122 Unfit

BTM2022-008 0.364 0.377 +0.013 Fit

BTM2022-009 0.364 0.367 +0.003 Fit

BTM2022-010 0.364 0.368 +0.004 Fit

39

3.3 Equipment

3.3.1 Testing apparatus

The testing apparatus used to perform the experimental tests of the present work was a

Biaxial Testing Machine (BTM), developed by Instituto Superior Técnico (IST) in collaboration with

Instituto Politécnico de Setúbal (IPS). This machine (Figure 3.3) was purposefully designed and built

aiming to test small samples of engineering materials in a very efficient manner and provide a low cost

alternative to the similar options available commercially. Figure 3.3 shows the cruciform arrangement

of the machine with the required 4 actuators, which are necessary to ensure symmetry and also to at

least minimise movement of the specimen centre during the test.

Figure 3.3 – Biaxial Testing Machine used to perform the experimental tests, [69].

The machine is composed by four iron-core linear motors with no mechanical devices to

produce linear movement, providing a fast response and zero backlash movement, [69]. The

assembly of the driving system, including iron-core motors, consists of an aluminium structure with a

guiding system without contact which is the result of the combination between air bearings and lateral

guides with rollers, making the system fit to sustain the constant attractive forces between the coil and

magnet, [69]. The magnetic forces developed between coil and magnet, have the purpose of pre-

loading the air bearings on the sides of the coil, allowing planar movement of the moving part

practically without friction, [69]. Figure 3.4 illustrates the assembly of the driving system.

Figure 3.4 – Driving system assembly, [69].

40

The BTM is assembled on a steel table with 2200 x 2200 mm size which weighs around 400

kg. The weight of each motor is around 80 kg, making the whole assembly over 700 kg. The

specifications of interest of the BTM are presented in Table 3.6, [69].

Table 3.6 – BTM specifications

Parameter Value/range

Maximum static force, [kN] ± 2.2

Maximum dynamic force, [kN] ± 3.5

Maximum test frequency, [Hz] 100

Moving mass, [kg] 32

Force transducers, [kN] ± 5

Encoder resolutions, [µm] 1

Maximum specimen length, [mm] 2000

Maximum displacement of each motor, [mm] 85

Table dimensions, [mm3] 2200 x 2200 x 1100

3.3.2 USB Microscope

In order to capture images of the centre of the specimen during the experimental tests, a Veho

VMS-001 200X USB microscope was employed. This device, worked in tandem with the interface

software of the BTM taking a picture on pre-defined intervals defined by the user. The microscope

proved to be a crucial element due to the fact that it allowed gathering important information regarding

crack propagation.

3.4 Experimental Methods

Ahead of every experimental test, a detailed procedure was carefully followed to ensure the

proper operation of the apparatus. The steps of the procedure consisted of setting machine zeros

properly, checking and correcting (if necessary) the alignment of the grips, assembly of the specimen

with the grips and setting of the test parameters.

The experimental tests with the apparatus and specimens previously described, were

conducted at room temperature, under load control, with constant load amplitude, equal loads on both

directions, stress ratio R=-1, and a frequency of 20 Hz, which provided stable behaviour of the

apparatus. A brief description of the parameters that differed between tests is presented in Table 3.7.

The phase parameter was always set to the second direction.

41

Table 3.7 – Test parameters for each specimen

Specimen ID Load amplitude, [kN] Phase shift, [°]

BTM2022-003 2.7 0

BTM2022-004 2.0 0

BTM2022-005 2.2 0

BTM2022-008 1.7 45

BTM2022-009 1.2 90

BTM2022-010 0.65 180

Taking the previous table into account, the loading waveforms were defined as sine waves of

the following form:

𝐹1 = 𝐹𝑎sin (𝜔𝑡) (3.1)

𝐹2 = 𝐹𝑎sin (𝜔𝑡 + 𝛿) (3.2)

Where 𝐹𝑎 is the load amplitude, 𝐹1 and 𝐹2 are the loads in directions 1 and 2 respectively, 𝜔 is

the frequency, 𝑡 is the time and 𝛿 is the phase.

As a consequence of using loadings with phase shifts, the loading paths lose the linear

behaviour, for a more complex one. In Figure 3.5 the variation of the loads during one complete cycle

and in Figure 3.6 the corresponding loading path can be observed.

(a) (b)

(c) (d)

Figure 3.5 – Variation of loads during a complete cycle: (a) In-phase loading; (b) phase shift of 45°; (c) phase shift of 90°; (d) phase shift of 180°.

42

(a) (b)

(c) (d)

Figure 3.6 – Resulting load paths: (a) In-phase loading; (b) phase shift of 45°; (c) phase shift of 90°; (d) phase shift of 180°.

The variations and load paths depicted in Figure 3.5 and Figure 3.6 are simply illustrative with

load amplitude of 1kN and time duration representing the duration of a single cycle.

The tests were allowed to take place until fracture of the specimen occurred and crack

initiation and propagation were monitored by the USB microscope mentioned in subsection 3.3.2.

43

4 Numeric Study

The numeric study performed in the development of this work, was based on a finite element

analysis (FEA) recurring to commercial FEA code ABAQUS®, which took into account the existence of

a crack and allowed computing the SIF for the different crack sizes obtained at different instants from

the monitoring of crack growth during the experimental tests. Due to this, the study described in this

chapter was only conducted after the experimental tests.

4.1 Specimen and crack modelling

In order to perform the analysis, the specimen model was created with half thickness, to

simplify the modelling and also facilitate the procedure to create the crack in the specimen. Figure 4.1

shows the model created.

Figure 4.1 – Specimen model in ABAQUS®.

To be able to account for the existence of a crack, the crack must be modelled considering

certain aspects that allow the analysis to simulate the theoretical singularity in stress at the ideal crack

tip, shown in Figure 4.2.

Figure 4.2 – Ideal and real crack tip comparison, [63].

In order to simulate the crack behaviour, the crack tip was modelled with small circles around

its end, partitioned into quarters as shown in Figure 4.3. Partitioning the circles is a step of elevated

44

importance to emulate the singularity at the crack tip. Due to the fact that ABAQUS® only recognizes

one crack tip, a second crack tip was modelled, following the same modus operandi.

Figure 4.3 – Crack tip model detail.

The centre of the specimen with the two cracks modelled is shown in Figure 4.4. The

dimensions indicated correspond to: a as the crack half-length, measured from the centre of the

specimen to the crack tip, and β is the initiation angle, as measured from the experimental tests,

relative to direction 2 (as identified in Figure 3.1).

Figure 4.4 – Close-up of both cracks modelled in ABAQUS®.

The definition of the crack through the options available in the software [70], was done in the

interaction module and two cracks were created as only one crack tip is recognized per crack. The

type of crack to be computed was set as a contour integral and the virtual growth direction was

defined with a q vector along the crack plane. The singularity was also characterized in this stage by

defining duplicate nodes at the crack tips as well as collapsed element sides and also setting the mid-

side node parameter to ¼ which moved the nearest nodes to a position closer to the crack tip.

45

4.2 Mesh and element type

In the interest of modelling the singularity at the crack tip and providing a regular mesh, a

different element type must be applied to the four quarters right at the end of the crack tip. The

quarters were seeded for ten elements each while the straight lines that divide the circle into quarters

are seeded for a single element. The remainder of the mesh was seeded in a way that yielded a

regular mesh and transitions as smooth as possible, establishing a compromise between results and

computational time.

The seeding was defined in the aforementioned way, to enable using wedge elements in the

crack tip region. The wedge elements used for this region were triangular prisms with 15 nodes with

the internal ABAQUS® designation C3D15 (Figure 4.5a), while the remaining elements used in the

mesh were hexahedral bricks with 20 nodes and reduced integration, designated by ABAQUS® as

C3D20R (Figure 4.5b). Both element types were 3D solids and of quadratic geometric order.

(a) (b)

Figure 4.5 – Element types used for the specimen model; (a) C3D15 wedge element; (b) C3D20R brick element, [71].

The resulting mesh was constituted by 32856 elements and 154172 nodes. Figure 4.6 shows

the detail at the crack tip, where the presence of the wedge elements is noticeable, and a

representative sample of the rest of the mesh, showing the regularity of the mesh.

(a) (b)

Figure 4.6 – Specimen mesh (a) Crack tip detail; (b) Sample of the rest of the mesh.

46

The half-thickness of the specimen was discretised with 6 elements, allowing ABAQUS® to

compute the J-Integral in 13 points through the modelled thickness and along the crack front.

4.3 Boundary Conditions and Loads

As usual for FEA to take place, both boundary conditions and loads are required. In the

present case, both of these parameters were set in a way to provide a reasonable approximation of

the real testing conditions.

A condition of high importance that was required is related to the centre of the specimen,

which should not move during the test. In order to simulate this behaviour in the FEA, displacement

boundary conditions were applied to the faces along the specimen’s thickness as shown in Figure 4.7.

The aforementioned conditions consisted of constraining displacement in the direction normal to the

faces identified in Figure 4.7. It should be noted that the reference frame is included to demonstrate to

which direction (x, y or z) the condition is respective, and also only two faces were constrained this

way since the remaining two were used to apply the loads as will be described ahead. For both faces,

the displacement was set to zero, allowing the specimen to move in the direction normal to the one

constrained at that face.

(a) (b)

Figure 4.7 – Displacement boundary conditions on the specimens extremities. (a) along the x direction; (b) along the y direction.

To complete the boundary conditions, a symmetry condition was applied to the “flat” face of

the specimen. This symmetry condition consisted of constraining three degrees of freedom: a

displacement constraint along the z direction, a rotation constraint around x and another rotation

constraint around y. All of the three constraints mentioned were given a value of zero.

With the boundary conditions defined, the loads were applied in the same way as the

displacement boundary conditions along the x and y directions, however, the loads were applied on

the opposite faces. In order to account for the effect of the combination of loads during a complete

cycle, the loads were discretized for a single step (corresponding to a complete loading cycle), with

increments of 0.05 and using equations (3.1) and (3.2) (section 3.4). The loads were applied as a

uniform pressure on the faces previously mentioned, and the amplitude was defined as the load

47

applied on the respective face, divided by the area. Figure 4.8 shows the faces and to which the loads

were applied as well as the axis orientation.

(a) (b)

Figure 4.8 – Loads applied on the specimens extremities. (a) along the x direction; (b) along the y direction.

4.4 Theoretical concepts applied to the numeric study

In the interest of computing the SIF from these analyses, gathering of the necessary data must

be requested as a history output for each individual crack. For the present case, the computation of

the SIF was requested for every increment and with five contours. This allowed obtaining SIF values

during a loading cycle, and the requested contours were used to compute the mean value of the SIF

for the respective increment.

Due to the fact that the loading applied to the specimen provokes both mode I and II, an

equivalent SIF should be computed. In order to obtain the aforementioned equivalent SIF, the model

presented by Richard et al., [72] was employed. Considering the SIF values for mode I and II, the

equivalent SIF can be computed in the following way:

𝐾𝑒𝑞 =𝐾𝐼

2+

1

2√𝐾𝐼

2 + 4(1.155𝐾𝐼𝐼)2 (4.1)

Where 𝐾𝐼 and 𝐾𝐼𝐼 are the stress intensity factors for loading modes I and II respectively. The

equivalent SIF range was computed through equation (4.2).

∆𝐾𝑒𝑞 = 𝐾𝑒𝑞,𝑚𝑎𝑥 − 𝐾𝑒𝑞,𝑚𝑖𝑛 (4.2)

Where ∆𝐾𝑒𝑞 is the SIF range, 𝐾𝑒𝑞,𝑚𝑎𝑥 is the maximum value of the equivalent SIF for a

complete load cycle and 𝐾𝑒𝑞,𝑚𝑖𝑛 is the minimum value of the equivalent SIF for a complete load cycle.

As the SIF is dependent on crack geometry, stress range, crack dimension, the analysis

conducted in this numeric study was based on the results of the experimental tests, i. e. crack length

and the angle of the crack were used in the finite element modelling.

48

5 Results and Discussion

This chapter presents the results of the study conducted, overviewing the results from the

critical plane models, the results of the experimental tests, numeric study results and the respective

correlation with the experimental results.

5.1 Critical plane models

The results presented in this section aim to predict the crack initiation angles, even though

these contemplate only to the out-of-phase loadings due to the fact that the critical plane models take

into account the contribution of shear stress, which for the case of in-phase loadings is equal to zero,

therefore making the critical plane parameters constant for every angle. The model constants

considered, were based on the work conducted by Fonseca in his master thesis, [73].

5.1.1 Findley Model

The variation of the damage parameter of the Findley model is shown in Figure 5.1, where the

three phase shift load cases are represented. The theoretical formulation of this parameter is given in

section 2.2.7.1.

Figure 5.1 – Findley parameter variation.

The crack initiation angle prediction with the Findley model shows the same general trend for

all three out-of-phase loading paths. However, it should be noted that for the 180° case, there is a

slight decrease in the damage parameter value at -45° and 45°, which are the angles for which the

damage parameter is maximum for the 45° and 90° phase shift cases. The maximum damage

parameter for the case with 180° phase shift occurs at -50°, -40°, 40° and 50°.

49

It is worthy to mention that the maximum damage parameter for each load path is the same for

every respective occurrence.

5.1.2 Brown and Miller Model

The formulation of the Brown and Miller parameter is found in section 2.2.7.2. The variation of

the three out-of-phase loading paths is shown in Figure 5.2.

Figure 5.2 – Brown and Miller parameter variation.

The Brown and Miller model shows a similar behaviour of the damage parameter for the three

load paths represented. Although the behaviour of the parameter is identical for every load path, the

maximum values occur for slightly different angles. In the case of the 45° phase shift, the maximum

values occur at angles of -72°, -18°, 18° and 72°. For the 90° phase shift, the maximum damage

parameter is seen at angles of -74°,-16°, 16° and 74°. The load path with a phase shift of 180°, has

maximum damage parameters at angles of -75°, -15°, 15° and 75°.

Analogously to the previous model, the maximum damage parameters are equal for every

occurrence of the respective load path.

5.1.3 Fatemi and Socie Model

A brief explanation of the Fatemi and Socie model was given in section 2.2.7.3. Figure 5.3

shows the variation of the Fatemi and Socie damage parameter.

The behaviour of the Fatemi and Socie damage parameter is similar for all three load paths

presented. However, for the case of the 180° phase shift the maximums occur at angles slightly

different from the ones corresponding to the maximum damage parameter for both the 45° and 90°

50

phase shift loadings. Both 45° and 90° phase shift load paths have maximum damage parameters for

angles of -45° and 45°. In the remaining case, the maximum damage parameter is found at angles of -

55°, -35°, 35° and 55°.

Figure 5.3 – Fatemi and Socie parameter variation.

5.1.4 Smith, Watson and Topper Model

The evolution of the SWT parameter is shown in Figure 5.4. The theoretical principle of this

model was described in section 2.2.7.4.

Figure 5.4 – Smith, Watson and Topper parameter variation.

51

This model displays a different behaviour than the ones presented previously, as in this model

the damage parameter for the case with a 180° phase shift, appears to be constant and equal to zero

in the spectrum of angles considered. However, the damage parameter for this case does vary

although it is with negligible values (10-17

), be that as it may, if the computations are performed for this

case, the behaviour is verified to be identical to the 45° and 90° phase shift cases. The maximum

values of damage parameter in this model were found at angles of -90°, 0° and 90°.

5.1.5 Liu I and II Model

As stated in section 2.2.7.5, the Liu model is separated into two parts each one related with a

failure mode, i.e. Mode I (opening) for Liu I and Mode II (sliding) for Liu II. The evolution of both these

parameters is shown in Figure 5.5 and Figure 5.6. Both Liu models are composed of a term related to

normal stress and strain and a term related to shear stress and strain.

Figure 5.5 – Liu I parameter variation.

Since the Liu I parameter defines the critical plane based on the maximum value of the

normal stress/strain term that is the quantity plotted in Figure 5.5, and it is possible to observe that for

all load cases, the maximum damage parameter is found at angles of -90°, 0° and 90°, which are all

angles normal to one of the loading directions of the specimen.

For the Liu II parameter, the situation is reversed in a sense. For this model, the critical plane

is defined based on the maximum value of the shear stress/strain term, therefore, that is the quantity

shown in Figure 5.6. In this case, the behaviour of the parameter is identical for the different load

paths and all maximum values can be found at angles of -45° and 45°. Although it might not be

noticeable, the damage parameter for the 180° and 45° share the same values, hence the curves

overlap each other in Figure 5.6.

52

Figure 5.6 – Liu II parameter variation.

5.1.6 Chu, Conle and Bonnen Model

The damage parameter of Chu, Conle and Bonnen was defined in section 2.2.7.6, and its

variation is shown in Figure 5.7. In this model the behaviour of the damage parameter is similar for all

loading cases, and the respective maximum values are found at angles of -45° and 45°.

Figure 5.7 – Chu, Conle and Bonnen parameter variation.

53

5.2 Experimental results

5.2.1 Crack Initiation

The results presented in this subsection show the images captured by the USB microscope

immediately before and when the crack was noticed, for each loading path tested. The angles at which

the crack initiated are also identified. For every case presented, the load directions are aligned with

the figures in the following way: 𝐹1 is along the vertical direction of the figure; 𝐹2 is along the horizontal

direction of the figure, which corresponds to the direction that was subjected to the phase-shift in the

respective cases.

The absence of specimen BTM2022-003 from this subsection is justified with the lack of fine-

tuning the cycle intervals for image capturing. The test of this specimen had an interval of 5000 cycles

between every capture. Due to this, the first image captured already showed a crack length of the

same order as the area of thickness reduction in the specimen. This specimen will, however, be

presented in the subsection dedicated to fracture surfaces.

For specimen BTM2022-004, the images before the crack was identified and when the crack

was identified are shown in Figure 5.8.

(a) (b)

Figure 5.8 – Specimen BTM2022-004; (a) at 453924 cycles; (b) at 456444 cycles.

The crack was noticed at 456444 cycles and the images were captured in intervals of

approximately 2500 cycles. The crack initiation angle is seen to be 6° relative to the direction of arms

2-4, however it must be clarified that the rolling direction of this specimen was not parallel with arms 1-

3 (and consequently not perpendicular to arms 2-4), instead the rolling direction showed an angle of

approximately 15°.

Specimen BTM2022-008, was tested with a phase shift in loading of 45°. The images

captured before the crack was identified and at the time the crack was identified are shown in Figure

5.9. For this test, the image capture was set to occur in intervals of 1000 cycles, gathering more

information regarding crack propagation, and initiation at an early stage. The rolling direction was as

intended for this specimen, i. e. parallel to arms 1-3, perpendicular to arms 2-4. The crack initiated

with a small angle, which was measured as 1°, however it is not represented in Figure 5.9.

54

The same situation for specimen BTM2022-009 is presented in Figure 5.10. This specimen

was also tested with out-of-phase loading, with a phase shift of 90°

Images were captured in intervals of 1000 cycles and the crack initiated at 0°, relative to the

direction of arms 2-4.

Figure 5.11 shows the images captured before and at the time the crack was detected for

specimen BTM2022-010. This test was performed with a 180° out-of-phase loading.

(a) (b)


(a) (b)


(a) (b)


55

The crack initiated with an angle of 2°, relative to the direction of arms 2-4, which is not

represented in Figure 5.11.

A comparative overview of the theoretical results for the critical plane and the experimental

results regarding crack initiation is presented in Table 5.1. It should be noted that, as referred in

section 5.1, the critical plane models were not used to predict crack initiation. Specimens are identified

only by the respective identification number.

Table 5.1 – Comparative overview of theoretical and experimental results for crack initiation.

Findley BM FS SWT Liu I Liu II Chu Experimental

004 - - - - - - - 6°

005 - - - - - - - 6°

008 ±45° ±72°/±18° ±45° ±90°/0° ±90°/0° ±45° ±45° 1°

009 ±45° ±74°/±16° ±45° ±90°/0° ±90°/0° ±45° ±45° 0°

010 ±50°/±40° ±75°/±15° ±55°/±35° ±90°/0° ±90°/0° ±45° ±45° 2°

From the analysis of Table 5.1, it is possible to verify that SWT and Liu I are the models that

are closer to the experimental results. These models define the critical plane based only on normal

strain/stress. When cracks initiate with a near zero angle a situation of Mode I loading might arise, and

control crack propagation due to the fact that mode I is the most dangerous.

5.2.2 Crack Propagation

Following the crack initiation results, this subsection presents the experimental data of crack

length vs number of cycles as well as some of the images captured for all specimens.

The relation between crack length and the corresponding number of cycles at which it was

verified for the test of specimen 004 is shown in Figure 5.12.

Figure 5.12 – Crack length vs number of cycles for specimen 004.

56

The images captured during the test of specimen 004 are shown in Figure 5.13.



(a) (b)

(c) (d)

Figure 5.13 – Crack propagation of specimen 004; (a) at 458964 cycles; (b) at 461484 cycles; (c) at 479117 cycles; (d) at final fracture (480826 cycles).


57

The images captured during the test of specimen 005 are shown in Figure 5.15.

(a) (b)

(c) (d)


Specimens 004 and 005 were both tested with the same load path and as seen in the

previous section, cracks initiated with the same angle. From the analysis of Figure 5.13 and Figure

5.15 it is possible to note the cracks propagated approximately along the direction of initiation.

For both these specimens, the direction of rolling of the sheet was of approximately 15°

relative to the vertical direction of the images. Although the applied loads were quite different and the

influence of that on specimen life is very significant, both specimens presented the same behaviour

regarding crack propagation, which in this case translates as the crack propagation occurring nearly

transverse to the direction of rolling of the sheet.


verified for the test of specimen 008 is shown in Figure 5.16. The images captured during the test of

specimen 008 are shown in Figure 5.17.

58


(a) (b)

(c) (d)

Figure 5.17 – Crack propagation of specimen 008; (a) at 41959 cycles; (b) at 43999 cycles; (c) at 52156 cycles; (d) at 62332 cycles.

The test of specimen 008 was conducted with an out-of-phase loading, with the phase shift set

at 45°. This generated an elliptical load path, as shown in section 3.4. The rolling direction of the sheet

59

was aligned with the directions of the specimen, and in Figure 5.17 the rolling direction is in the

vertical direction. The crack propagation behaviour of specimen 008 resembles of the behaviour for

the in-phase loading tests to a certain extent. Cracks propagated along the same direction as they

initiated. However, branching occurred at a certain point (Figure 5.17c), perpendicular to the main

crack. This phenomenon was noticed again around 10000 cycles later (Figure 5.17d).

Even though this was an out-of-phase loading, the main crack initiated and propagated in a

direction approximately normal to the sheet rolling direction, showing similar behaviour to the in-phase

loading specimens.




The images captured during the test of specimen 009 are shown in Figure 5.19. Specimen

009 was also subjected to out-of-phase loading with the phase shift set to 90°. The rolling direction is

in the vertical direction of the images captured. The crack propagation behaviour shown in this test is

along the same trend as previous experimental tests. The crack propagated along the same direction

as it initiated. In the same manner as specimen 008, specimen 009 also showed crack branching

(Figure 5.19c), slightly different than the configuration showed by specimen 008.

Even though, significant branching was verified, the main crack still propagated in a direction

normal to the rolling direction of the sheet, maintaining the tendency of the previous tests.


verified for the test of specimen 010 is shown in Figure 5.20. The images captured during the test of

specimen 010 are shown in Figure 5.21.

60

(a) (b)

(c) (d)



61

(a) (b)

(c) (d)


The loading path for specimen 010 was fully reversed, i.e. 180° out-of-phase. This means that

at the point of maximum load for one direction, there is maximum compression in the perpendicular

direction, contributing to the tensile stress at the centre of the specimen in a considerably different

way, than the previous load paths. Despite that, crack propagation behaviour followed the trend set by

the previous tests: crack propagated in the direction of initiation, which is normal to the sheet rolling

direction. In similar fashion to the other out-of-phase tests, crack branching occurred, with the

branches forming, perpendicular to the main crack.

5.3 Numeric study results

In this subsection, the results of the numeric study are presented. The numeric analyses were

performed to determine the SIF for mode I and II, and compute the equivalent SIF as explained in

section 4.4. In addition, the stress distribution at the crack tip is shown for the increment in which the

crack is open.

5.3.1 Specimen 004 and 005

The numeric results for specimens 004 and 005 are presented together due to the fact that

both specimens were subjected to in-phase load paths. The stress distribution at the crack front for

specimen 004 is presented in Figure 5.22.

62

The equivalent SIF range, (∆𝐾𝑒𝑞), for each measurement of crack length obtained is shown in

Table 5.2. It should be noted that only half the crack length, (𝑎), is presented in accordance to the

finite element model. The stress distribution at the crack front for specimen 005 is presented in Figure

5.23. The equivalent SIF range for each measurement of crack length obtained is shown in Table 5.3.

Figure 5.22 – Stress distribution at crack tip for specimen 004.

Table 5.2 – Equivalent SIF range relation with respective half crack length for specimen 004

𝑎, [mm] 𝐾𝐼, [MPa√m] 𝐾𝐼𝐼, [MPa√m] ∆𝐾𝑒𝑞, [MPa√m]

1.025 16.14 0.06 16.14

1.940 18.09 0.16 18.10

3.000 18.40 0.26 18.41


63



0.325 10.67 0.04 10.67

0.790 16.02 0.06 16.02

1.140 16.02 0.06 16.02

1.485 19.46 0.02 19.46

2.445 19.39 0.26 19.40

For both cases the stress distribution displays the typical behaviour of the plastic zone at the

crack front. Also, it is possible to observe that the equivalent SIF range increases with crack length,

except for the last crack length of specimen 5, which is related to the fact that the increasing crack

length leads to more plastic deformation and plasticity at the crack tip, which in turn goes against

using an elastic parameter like the stress intensity factor range.

5.3.2 Specimen 008

The stress distribution at the crack front for specimen 008 is presented in Figure 5.24.

Although specimen 008 was subjected to an out-of-phase loading, at the increment where the

crack is open the stress distribution still displays the usual behaviour of the plastic zone at the crack

front.

The equivalent SIF range for each measurement of crack length obtained is shown in Table

5.4.


64



0.135 6.26 0.17 6.26

0.350 10.02 0.23 10.02

0.780 14.42 0.31 14.42

1.190 17.06 0.32 17.06

1.710 18.12 0.36 18.12

2.060 18.73 0.45 18.73

2.585 18.24 0.41 18.25

2.965 17.38 0.33 17.38

As seen for specimen 005, the equivalent SIF range seems to drop after a certain point, which

again is related to the fact that crack propagation is unstable, due to the large crack size resulting in a

significant stiffness decrease of the specimen.

5.3.3 Specimen 009


Specimen 009 also displays the expected stress distribution of the plastic zone at the crack

front. The equivalent SIF range for each measurement of crack length obtained is shown in Table 5.5.


The trend of unstable crack propagation indicated by the reduction of the equivalent SIF range

is verified for this case too.

65

Table 5.5 – Equivalent SIF range relation with respective half crack length for specimen 009.


0.095 4.78 0.03 4.78

0.150 6.17 0.03 6.17

0.260 7.94 0.004 7.94

0.365 9.50 0.009 9.50

0.600 12.19 0.005 12.19

0.800 13.59 0.009 13.59

0.920 14.29 0.01 14.29

1.460 16.46 0.09 16.47

2.130 17.82 0.002 17.82

3.100 16.58 0.005 16.58

3.465 16.45 0.03 16.45

5.3.4 Specimen 010


Even for the fully reversed load path, the stress distribution at the crack front, displays the

typical behaviour at the crack front.

The equivalent SIF range for each measurement of crack length obtained is shown in Table

5.6.


66

The evolution of the equivalent SIF range for specimen 010 shows a behaviour slightly

different than the trend obtained so far, with the increase in equivalent SIF range having a couple of

points with opposite (decrease) effect.

Table 5.6 – Equivalent SIF range relation with respective half crack length for specimen 010.


0.325 6.03 0.44 6.07

0.565 7.96 0.55 8.01

0.790 9.24 0.67 9.31

1.040 10.27 0.76 10.35

1.135 10.23 0.77 10.31

1.310 10.72 0.87 10.81

1.795 11.65 0.81 11.72

2.130 11.87 0.88 11.95

2.320 12.08 0.97 12.18

2.565 12.09 0.85 12.18

2.955 12.08 0.80 12.15

3.210 12.18 0.78 12.24

3.565 11.19 0.80 11.27

5.4 Correlation of experimental and numeric data

The aim of the numeric study was to compute the equivalent stress intensity factor range in

order to correlate it with the crack propagation rate obtained from the experimental tests, and obtain

the power law constants, i.e. establish the Paris Law equation. The images of the test of specimen

004, did not capture early crack length and therefore, the correlation between numerical and

experimental results was not developed for this specimen.

For every data relation in this section, the crack growth rate was computed through the

variation in crack size during the corresponding cycle interval. All plots are represented in log-log

scales in order to simplify reading.

The plot corresponding to both the numeric results and the experimental results for specimen

005 is presented in Figure 5.27. The power trend line added to the data set, provides a relation of the

same form as the Paris Law (equation (2.1), section 2.2.4), and for this particular case the constants

were given with the following values: 𝐶 = 1 × 10−11 (mm/cycle)/(MPa√m)m and 𝑚 = 5.82 and with a

goodness of fit of 𝑅2 = 0.8634. The constants found with this trend line show some agreeance with the

results found in literature, namely references [74] and [75] , even though other magnesium alloys were

the object of study.

67

Figure 5.28 shows the relation between the crack growth rate and the equivalent SIF range for

specimen 008, which was subjected to an out-of-phase loading path, with a phase shift of 45°.

Figure 5.27 – da/dN vs ΔKeq


68

The power trend line obtained in this case yielded the following constants: 𝐶 = 6 × 10−9

(mm/cycle)/(MPa√m)m and 𝑚 = 3.93 with a goodness of fit 𝑅2 = 0.8247. The constants obtained are

along the same order of magnitude as the constants given in references [76] and [77].

The plot of the crack propagation vs equivalent SIF range of specimen 009, which was

subjected to a 90° out-of-phase loading, is presented in Figure 5.29.


The power trend line obtained in this case yielded the following constants: 𝐶 = 2 × 10−9

(mm/cycle)/(MPa√m)m and 𝑚 = 4.45 with a goodness of fit 𝑅2 = 0.7172. The constants obtained are

along the same order of magnitude as the constants given in references [74] and [77].

Figure 5.30 shows the relation between crack growth rate and the equivalent SIF range for

specimen 010, which was subjected to a fully reversed loading cycle.

For this particular case the data obtained showed larger scatter, with a lower goodness of fit

than in previous cases (𝑅2 = 0.6637). The constants obtained with the power trend line added to the

data set were the following: 𝐶 = 4 × 10−12 (mm/cycle)/(MPa√m)m and 𝑚 = 7.4. No correspondence to

the constants obtained was found in the literature, particularly for the value of m.

69


5.5 Fracture surface analysis

In order to observe and understand the effects of in-plane biaxial loading crack growth, some

fracture surfaces are presented in this section.

In Figure 5.31 it is possible to observe indicators of crack propagation like the smoother

surface in Figure 5.31a followed by a region with a rougher surface, which indicate that the crack

propagated due to cyclic loading rather than overload, up to the point where material strength

decreased enough, leading to final fracture.

Another indication is present in Figure 5.31b where it is possible to observe radial marks

fanning out from the crack initiation site suggesting rapid crack growth took place, which is coherent

with the numeric/experimental data correlations, in the way that the exponents found for the crack

propagation power laws consisted of considerably large values.

Figure 5.32 presents two separate ends of specimen 004. It is possible to observe that a

secondary crack formed (Figure 5.32a), which jointly with the reduction in strength due to the

formation of the cracks, may have led to buckling as seen in Figure 5.32b by the end of the test.

70

(a) (b)

Figure 5.31 – Fracture surfaces of specimen 003.

(a) (b)


The shinier polished surface identified in Figure 5.32 suggests brittle cracking by cleavage or

intergranular fracture, it also indicates the surfaces were cyclically set against each other leading to

the polished surface effect.

71

The fracture surfaces of specimen 008, subjected to a 45° out-of-phase loading, are shown in

Figure 5.33. Once again it is possible to observe the radial marks fanning out of the crack initiation

site, and it is also possible to observe the growing surface roughness of the surface on Figure 5.33b

(a) (b)


72

6 Conclusions and Future Developments

The sixth and final chapter presents the conclusions drawn from this study as well as present

topics for future development, regarding testing of the material in the same apparatus.

6.1 Conclusions

Most critical plane models did not provide reasonable estimations for the crack

initiation angles, and no estimation in the cases of in-phase loading, due to the fact

that no shear stresses are explicitly applied;

Critical plane models of SWT and Liu I provided good estimations for crack initiation,

due to the fact that these models define the critical plane based on normal strains or

stresses;

Crack initiation and propagation nearly perpendicular to one of the loading directions

suggests heavier influence of mode I loading, the same way it was verified with the

numeric study;

Although the in-phase loading tests were performed with specimens that were not

aligned with the rolling direction, the general trend of crack initiation and propagation

normal to the rolling direction was achieved;

Crack branching leading to significant secondary cracks normal to the initial crack took

place for every out-of-phase case, which generated two mode I and mode II loadings

due to the geometry of the specimen and loading conditions;

The correlation of experimental and numerical data provided acceptable results,

namely the crack propagation power law constants, except for the fully reversed

loading cycle;

Further tests should be carried out in order to verify and consolidate the obtained

results.

6.2 Future Developments

As a final note, a few suggestions for future developments are given:

Further the experimental tests investigation with different load ratios, and presence of

mean stress to understand its effect;

Investigate the effect of different loading frequencies between the loading arms;

Define loading blocks to mimic a real load spectrum, with variable load amplitudes and

implement it in testing;

Investigate anisotropy effects with specimens obtained with different angles relative to

the sheet rolling direction.

73

References

[1] R. I. Stephens, A. Fatemi, R. R. Stephens and H. O. Fuchs, Metal Fatigue in Engineering,

John Wiley & Sons, Inc., 2001.

[2] “Norskolje&gass,” 13 December 2010. [Online]. Available:

https://www.norskoljeoggass.no/en/Facts/Petroleum-history/. [Accessed 20 February

2016].

[3] M. Bak, “Industrial Heating,” 7 January 2016. [Online]. Available:

http://www.industrialheating.com/articles/92637-failure-analysis-to-discovermitigate-

disastrous-crack-propagation. [Accessed 20 February 2016].

[4] D. A. Kramer, “Magnesium, its Alloys and Compounds,” [Online]. Available:

http://pubs.usgs.gov/of/2001/of01-341/of01-341.pdf. [Accessed 20 April 2015].

[5] H. E. Friedrich and B. L. Mordike, Magnesium Technology - Metallurgy, Design Data,

Applications, Berlin [etc.]: Springer, 2004.

[6] “International Magnesium Association - Where is Magnesium Found?,” [Online]. Available:

http://www.intlmag.org/magnesiumbasics/where.cfm. [Accessed 19 April 2015].

[7] M. O. Pekguleryuz, K. U. Kainer and A. A. Kaya, Fundamentals of Magnesium Alloy

Metallurgy, Cambridge: Woodhead Publishing, 2013.

[8] “The Essential Chemical Industry,” [Online]. Available:

http://www.essentialchemicalindustry.org/metals/magnesium.html. [Accessed 21 April

2015].

[9] E. A. Brandes and G. B. Brook, Smithells Light Metals Handbook, Oxford: Butterworth

Heinemann, 1988.

[10] “International Magnesium Association - Alloy Designation,” [Online]. Available:

http://www.intlmag.org/magnesiumresources/designation.cfm. [Accessed 17 April 2015].

[11] “Total Materia,” [Online]. Available:

http://www.totalmateria.com/page.aspx?ID=CheckArticle&LN=PT&site=ktn&NM=4.

[Accessed 17 April 2015].

[12] “UCDavis ChemWiki,” [Online]. Available:

http://chemwiki.ucdavis.edu/Physical_Chemistry/Physical_Properties_of_Matter/Phases_of

_Matter/Solids/Crystal_Lattice/Closest_Pack_Structures. [Accessed 2015].

74

[13] M. M. Avedesian and H. Baker, ASM Specialty Handbook: Magnesium and Magnesium

Alloys, ASM International, 1999.

[14] “Fire Safety Advice Centre,” [Online]. Available: http://www.firesafe.org.uk/portable-fire-

extinguisher-general/. [Accessed 22 April 2015].

[15] I. Ostrovsky and Y. Henn, “Present State and Future of Magnesium Applications in

Aerospace Industry,” in New Challenges in Aeronautics, Moscow, 2007.

[16] “International Magnesium Association - Applications,” [Online]. Available:

http://www.intlmag.org/magnesiumapps/overview.cfm. [Accessed 22 April 2015].

[17] “Sikorsky Archives,” [Online]. Available: http://www.sikorskyarchives.com/S+56%20HR2S-

1H-37.php. [Accessed 30 April 2015].

[18] “National Museum of the US Air Force,” [Online]. Available:

http://www.nationalmuseum.af.mil/factsheets/factsheet.asp?id=290. [Accessed 30 April

2015].

[19] “Military Factory,” [Online]. Available: http://www.militaryfactory.com/imageviewer/ac/pic-

detail.asp?aircraft_id=235&sCurrentPic=pic1. [Accessed 30 April 2015].

[20] “Air Vectors,” [Online]. Available: http://www.airvectors.net/avbear.html. [Accessed 30 April

2015].

[21] A. Kenger and G. Leavy, “Development and Fabrication of the F-80C Magnesium Alloy

Airframe,” East Coast Aeronautics, Inc., 1955.

[22] “www.joebaugher.com,” [Online]. Available:

http://www.joebaugher.com/usaf_fighters/p80_6.html. [Accessed 30 April 2015].

[23] “Magnesium Solutions,” [Online]. Available:

http://www.magnesiumsolutions.com/index.html. [Accessed 28 April 2015].

[24] “Bold Ride,” [Online]. Available: http://www.boldride.com/ride/1938/volkswagen-beetle.

[Accessed 5 May 2015].

[25] “GF Automotive,” [Online]. Available: http://www.gfau.com/content/gfau/com/en/products-

and-solutions/passenger-car.html. [Accessed 7 May 2015].

[26] F. I. d. l'Automobile, 2015 Formula One Technical Regulations, Fédération Internationale

de l'Automobile, 2014.

[27] W. Schütz, “A History of Fatigue,” Engineering Fracture Mechanics, vol. 54, pp. 263-300,

1996.

75

[28] Y. S. Garud, “Multiaxial Fatigue: A Survey of the State of the Art,” Journal of Testing and

Evaluation, vol. 9, 1981.

[29] D. F. Socie and G. B. Marquis, Multiaxial Fatigue, Warrendale, PA: Society of Automotive

Engineers, 2000.

[30] J. J. Blass and S. Y. Zamrik, “Multiaxial Low-Cycle Fatigue of Type 304 Stainless Steel,”

Oak Ridge, Tennessee, 1976.

[31] K. Kanazawa, K. J. Miller, and M. W. Brown, “Low-Cycle Fatigue Under Out-of-Phase

Loading Conditions,” Journal of Engineering Materials and Technology, vol. 99 (3), pp.

222-228, 1977.

[32] K. Kanazawa, K. J. Miller and M. W. Brown, “Cyclic Deformation of 1% Cr-Mo-V Steel

Under Out-of-Phase Loads,” Fatigue of Engineering Materials and Structures, vol. 2, pp.

217-228, 1979.

[33] A. Fatemi and D. F. Socie, “A Critical Plane Approach to Multiaxial Fatigue Damage

Including Out-of-Phase Loading,” Fatigue & Fracture of Engineering Materials &

Structures, vol. 11 (3), pp. 149-165, 1988.

[34] A. Matos, “Multiaxial Fatigue Simulation of an AZ31 Magnesium Alloy,” 2010, MSc Thesis.

[35] I. V. Papadopoulos, “Critical Plane Approaches in High-Cycle Fatigue: On the Definition of

the Amplitude and Mean Value of the Shear Stress Acting on the Critical Plane,” Fatigue &

Fracture of Engineering Materials & Structures, vol. 21, pp. 269-285, 1998.

[36] M. Freitas, B. Li and J. Santos, “A Numerical Approach for High-Cycle Fatigue Life

Prediction with Multiaxial Loading,” in Multiaxial Fatigue and Deformation: Testing and

Prediction, ASTM, STP 1387, West Conshohocken, PA, American Society for Testing and

Materials, 2000, pp. 139-156.

[37] L. Reis, B. Li and M. Freitas, “Multiaxial Fatigue Testing and Analysis of Metallic

Materials,” Anales de Mecánica de la Fractura, vol. 20, pp. 462-467, 2003.

[38] L. Reis, “Comportamento Mecânico de Aços em Fadiga Multiaxial a Amplitude de Carga

Constante e Síncrona,” Lisbon, Portugal, 2004, PhD Thesis.

[39] Y.-Y. Wang and W.-X. Yao, “A Multiaxial Fatigue Criterion for Various Metallic Materials

under Proportional and Nonproportional Loading,” International Journal of Fatigue, vol. 28,

pp. 401-408, 2006.

[40] S. Hasegawa, Y. Tsuchida, H. Yano and M. Matsui, “Evaluation of Low Cycle Fatigue Life

in AZ31 Magnesium Alloy,” International Journal of Fatigue, vol. 29, pp. 1839-1845, 2007.

76

[41] M. Tsushida, K. Shikada, H. Kitahara, S. Ando and H. Tonda, “Relationship Between

Fatigue Strength and Grain Size in AZ31 Magnesium Alloys,” Materials Transactions, vol.

5, pp. 1157-1161, 2008.

[42] S. Begum, D. L. Chen, S. Xu and A. A. Luo, “Low cycle fatigue properties of an extruded

AZ31 magnesium alloy,” International Journal of Fatigue, vol. 31, pp. 726-735, 2009.

[43] K. Tokaji, M. Nakajima and Y. Uematsu, “Fatigue crack propagation and fracture

mechanisms of wrought magnesium alloys in different environments,” International Journal

of Fatigue, vol. 31, pp. 1137-1143, 2009.

[44] L. Reis, B. Li and M. Freitas, “Crack Initiation and growth path under multiaxial fatigue

loading in structural steels,” International Journal of Fatigue, vol. 31, pp. 1660-1668, 2009.

[45] J. D. Bernard, J. B. Jordon, M. F. Horstemeyer, H. El Kadiri, J. Baird, D. Lamb and A. A.

Luo, “Structure-property relations of cyfclic damage in a wrought magnesium alloy,” Scripta

Materialia, vol. 63, pp. 751-756, 2010.

[46] J. Albinmousa, H. Jahed and S. Lambert, “An Energy-Based Fatigue Model for Wrought

Mangnesium Alloy under Multiaxial Load,” in The Ninth International Conference on

Multiaxial Fatigue & Fracture (ICMFF9), Parma, 2010.

[47] L. Reis, B. Li and M. Freitas, “Mean stress effect during cyclic stress/straining of 42CrMo4

structural steel,” in The Ninth International Conference on Multiaxial Fatigue & Fracture

(ICMFF9), Parma, 2010.

[48] J. Albinmousa, H. Jahed and S. Lambert, “Cyclic behaviour of wrought magnesium alloy

under multiaxial load,” International Journal of Fatigue, vol. 33, pp. 1127-1139, 2011.

[49] J. Albinmousa, H. Jahed and S. Lambert, “Cyclic axial and cyclic torsional behaviour of

extruded AZ31B magnesium alloy,” International Journal of Fatigue, vol. 33, pp. 1403-

1416, 2011.

[50] R. Zeng, E. Han and W. Ke, “A critical discussion on influence of loading frequency on

fatigue crack propagation behavior for extruded Mg-Al-Zn alloys,” International Journal of

Fatigue, vol. 36, pp. 40-46, 2012.

[51] V. Anes, L. Reis, B. Li and M. Freitas, “AZ31 Magnesium alloy mechanical behaviour

under low-cycle-fatigue regime,” in XIII Portuguese Conference on Fracture, Coimbra,

2012.

[52] V. Anes, L. Reis, B. Li and M. Freitas, “Crack path evaluation on HC and BCC

microstructures under multiaxial cyclic loading,” International Journal of Fatigue, vol. 58,

77

pp. 102-113, 2014.

[53] V. Anes, L. Reis, B. Li, M. Fonte and M. Freitas, “New approach for analysis of complex

multiaxial loading paths,” International Journal of Fatigue, vol. 62, pp. 21-33, 2014.

[54] T. Itoh, M. Sakane and K. Ohsuga, “Multiaxial low cycle fatigue life under non-proportional

loading,” International Journal of Pressure Vessels and Piping, vol. 110, pp. 50-56, 2013.

[55] N. Shamsaei and A. Fatemi, “Small fatigue crack growth under multiaxial stresses,”

International Journal of Fatigue, vol. 58, pp. 126-135, 2014.

[56] R. Cláudio, L. Reis and M. Freitas, “Biaxial high-cycle fatigue life assessment of ductile

aluminium cruciform specimens,” Theoretical and Applied Fracture Mechanics, vol. 73, pp.

82-90, 2014.

[57] R. Baptista, R. Cláudio, L. Reis, I. Guelho, M. Freitas and J. F. A. Madeira, “Design

optimization of cuciform specimens for biaxial fatigue loading,” Frattura ed Integritá

Strutturale, vol. 30, pp. 118-126, 2014.

[58] V. Anes, L. Reis, B. Li and M. Freitas, “New approach to evaluate non-proportionality in

mutiaxial loading conditions,” Fatigue & Fracture of Engineering Materials & Structures,

vol. 37, pp. 1338-1354, 2014.

[59] V. Anes, L. Reis, B. Li, M. Freitas and C. M. Sonsino, “Minimum Circumscribed Ellipse

(MCE) and Stress Scale Factor (SSF) criteria for multiaxial fatigue life assessment,”

Theoretical and Applied Fracture Mechanics, vol. 73, pp. 109-119, 2014.

[60] V. Anes, L. Reis, B. Li and M. Freitas, “New cycle counting method for multiaxial fatigue,”

International Journal of Fatigue, vol. 67, pp. 78-94, 2014.

[61] V. Anes, L. Reis and M. Freitas, “Multiaxial Fatigue Damage Accumulation under Variable

Amplitude Loading Conditions,” Procedia Engineering, vol. 101, pp. 117-125, 2015.

[62] B. C. Li, C. Jiang, X. Han and Y. Li, “A new approach of fatigue life prediction for metallic

materials under multiaxial loading,” International Journal of Fatigue, vol. 78, pp. 1-10,

2015.

[63] N. E. Dowling, Mechanical Behavior of Materials: Engineering Methods for Deformation,

Fracture, and Fatigue, Pearson Education Limited, 2013.

[64] A. S. Ribeiro and A. M. P. Jesus, “Fatigue Behaviour of Welded Joints Made of 6061-T651

Aluminium Alloy,” in Aluminium Allots, Theory and Applications, InTech, 2011, p. 400.

[65] C. van Kranenburg, “Fatigue crack growth in Aluminium Alloys,” Delft, Holland, 2010, PhD

78

Thesis.

[66] S. Swift, “A Collective Approach to Aircraft Structural Maintenance Programs,” Civil

Aviation Safety Authority (CASA), Australia, 2008.

[67] “eFatigue,” [Online]. Available:

https://www.efatigue.com/multiaxial/background/strainlife.html. [Accessed 24 May 2015].

[68] “MatWeb - Material Property Data,” [Online]. Available:

http://www.matweb.com/search/DataSheet.aspx?MatGUID=d1e286e1ac0742358544b953

bbf3c2e9&ckck=1. [Accessed 12 April 2015].

[69] M. Freitas, L. Reis, B. Li, I. Guelho, V. Antunes, J. Maia and R. A. Cláudio, “In-Plane

Biaxial Fatigue Testing Machine Powered by Linear Iron-Core Motors,” Application of

Automation Technology in Fatigue and Fracture Testing and Analysis, vol. 6th Volume, pp.

63-79, 2014.

[70] Abaqus/CAE User's Guide.

[71] Abaqus Analysis User's Guide.

[72] H. A. Richard, M. Fulland and M. Sander, “Theoretical crack path prediction,” Fatigue &

Fracture of Engineering Materials & Structures, vol. 28, pp. 3-12, 2005.

[73] H. Fonseca, “Caracterização e avaliação do comportamento mecânico da liga de

Magnésio AZ31B sob fadiga multiaxial na presença de um entalhe,” Lisbon, Portugal,

2013, MSc Thesis.

[74] M. Mehrzadi, “Fatigue characterization of AM60B Magnesium alloy subjected to constant

and variable amplitude loading with positive and negative stress ratios,” Halifax, Nova

Scotia, Canada, 2013, PhD Thesis.

[75] P. Venkateswaran, S. G. S. Raman, S. D. Pathak, Y. Miyashita and Y. Mutoh, “Fatigue

crack growth behaviour of a die-cast magnesium alloy AZ91D,” Materials Letters, vol. 58,

pp. 2525-2529, 2004.

[76] K. U. Kainer, “Fatigue Life Prediction of Magnesium Alloys for Structural Applications,” in

Magnesium: Proceedings of the 6th International Conference on Magnesium Alloys and

Their Applications, 2004.

[77] R. C. Zeng, Y. B. Xu, W. Ke and E. H. Han, “Fatigue crack propagation behavior of an as-

extruded magnesium alloy AZ80,” Materials Science and Engineering A, vol. 509, pp. 1-7,

2009.

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Mechanical behaviour of AZ31B Magnesium alloy subjected to in … · iii Abstract The present work was carried out in order to better understand and characterize the mechanical behaviour