Continuum models of the mechanical behavior of rolled and die-cast ...

UNIVERSIDAD POLITECNICA DE MADRID

ESCUELA TECNICA SUPERIOR DE

INGENIEROS INFORMATICOS

Continuum models of the mechanical behavior of

rolled and die-cast magnesium alloys

TESIS DOCTORAL

ANA MARIA FERNANDEZ BLANCO

Ingeniero Tecnico Aeronautico

Master en Materiales Estructurales para las Nuevas Tecnologıas

2014

Doctorado en Computacion Avanzada

para Ciencias e Ingenierıas

Escuela Tecnica Superior de Ingenieros Informaticos

Universidad Politecnica de Madrid

PhD. Thesis

International Mention

Continuum models of the mechanical behavior of

rolled and die-cast magnesium alloys

Author

Ana Marıa Fernandez Blanco

BEng. in Aeronautical Technical Engineering

MEng. in Structural Materials for New Technologies

Supervised by

Dr. M. Teresa Perez-Prado

PhD. in Physics

Prof. Antoine Jerusalem

PhD. in Computational Mechanics of Materials

IMDEA Materials Institute, Madrid, Spain

June 2014

Tribunal nombrado por el Excmo. y Magfco. Sr. Rector de la U.P.M., el dıa .....

Presidente:

Secretario:

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Realizado el acto de defensa y lectura de la Tesis el dıa ..... de ........ del 2014 en la

E.T.S. de Ingenieros Informaticos de la U.P.M.

Calificacion: ...............

Firmado por el Tribunal Calificador:

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Vocales

Dedicated to my family

Acknowledgements

This thesis has been carried out at the IMDEA Materials Institute in Madrid and has

been defended at the Technical University of Madrid, Spain. It is a pleasure to have the

opportunity to thank all those who supported me during this work.

I am delighted to express my gratitude to my advisors Dr. M. Teresa Perez-Prado at

IMDEA Materials Institute and Prof. Antoine Jerusalem now at the University of Oxford.

I still remember our first meeting at IMDEA Materials Institute and the words they said

to me. Since that day, my life took a turn and I started enjoying the research field. Thanks

to their support, encouragement, excellent guidance and confidence in me have made it

possible to carry out this thesis. It has been an honor and a privilege to have had the

chance to work with them and learn from them. My appreciation is immense.

I gratefully acknowledge to Prof. Javier Llorca and Prof. Jose Manuel Torralba for

giving me the opportunity to complete my thesis at IMDEA Materials Institute, for their

support and for giving me sincere advices and valuable opportunities.

My most sincere thanks go to Dr. Federico Sket for sharing his knowledge in the field

of tomography which has been of great importance for this work. Thanks to him and

to Dr. Jon Molina-Aldareguia for their collaboration and for their inspiring, constructive

and critical discussions.

I would like to thank to Prof. Dr.-Ing. Dierk Raabe for giving me the opportunity to

perform my stay at Max-Planck-Institut fur Eisenforschung GmbH (MPIE) in Dusseldorf,

Germany. I am deeply grateful to Dr. Ivan Gutierrez-Urrutia for his help and fruitful

discussions during the stay at MPIE. Thanks are also due to Dr. Stefan Zaefferer, Dr.

Anahita Khorashadizadeh and Monika Nellessen for the technical support on the three-

dimensional electron backscatter diffraction measurements at MPIE.

I am sincerely grateful to Prof. Lallit Anand at the Massachusetts Institute of

Technology for supplying the original VUMAT for rate-independent crystal plasticity for

hexagonal close-packed materials written by Staroselsky and Anand. I would also like

to extend grateful thanks to Prof. Yujie Wei at State Key Lab of Nonlinear Mechanics

(Institute of Mechanics in Beijing), Prof. Ron W. Armstrong at the University of Maryland

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and Prof. Andy Godfrey at the Tsinghua University, for very useful, interesting and

productive discussions.

The technicians of the CAI X-ray diffraction of the Complutense University and the

National Center for Metals Research in Madrid are sincerely thanked for their kind

assistance. Thanks must also go to Hiperbaric S.A. for the help with the hydrostatic

pressure treatments. I would also like to acknowledge the use of the University of Oxford

Advanced Research Computing (ARC) facility in carrying out this work.

I thank to the vehicle interior manufacturer, Grupo Antolin Ingenierıa S.A., within

the framework of the project MAGNO2008-1028-CENIT Project funded by the Spanish

Ministry. Financial support from the PRI-PIBUS-2011-0917 project (MAGMAN) from

the Spanish Ministry of Economy and Competitiveness (MINECO) is also gratefully

acknowledged.

My thanks to all the members, past and present, at IMDEA Materials Institute for

the pleasant and warm atmosphere which exist in the institute. They are extraordinary

people. Despite the lack of space, I would like to mention some of them for their friendly

support: Carmen, Elena, Eva, Fede, Guillermo, Irene, Juan Pedro, Katia, Laura, Marcos,

Mariana, Paloma, Paqui, Rafa, Raul M., Raul S., Roberto, Rocıo S., Saeid, Sergio, Vanesa.

Special thanks to Nathamar for her invaluable help at the beginning of this thesis, to Rocıo

M. for her friendship and to Julian for his help and encouragement when I was feeling low.

I would like to thank my friends and my family. Many thanks to my parents and

brothers for all their love. They have taught me to be always persistent, optimistic and to

look forward to achieving even higher goals. And finally, to Jose, for all his patience and

love. He gave me his endless support and helped me to go through all the difficulties I

came across during this time. He provided me the required motivation to make this thesis

a reality.

To all of you, thanks!

Ana Fernandez

Contents

List of Abbreviations xi

Resumen xiii

Abstract xv

1 Introduction 1

1.1 Length scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Magnesium alloys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.3 Texture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.4 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2 State of the Art 11

2.1 Rolled Mg alloys: the influence of texture on the mechanical behavior . . . 11

2.2 Fundamentals of twinning . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.3 Die-cast Mg alloys: the influence of porosity on the mechanical behavior . . 14

2.4 Continuum models for HCP materials . . . . . . . . . . . . . . . . . . . . . 15

2.4.1 Crystal plasticity modeling . . . . . . . . . . . . . . . . . . . . . . . 15

2.4.2 Mechanical twinning in CPFE models . . . . . . . . . . . . . . . . . 16

2.4.3 Continuum models for porosity evolution . . . . . . . . . . . . . . . 17

2.5 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3 Experimental Campaign 19

3.1 Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.2 Microstructural characterization . . . . . . . . . . . . . . . . . . . . . . . . 19

3.2.1 Measurement of macrotexture by X-ray diffraction . . . . . . . . . . 20

3.2.2 Measurement of microtexture by electron backscatter diffraction . . 20

3.2.2.1 2D EBSD technique . . . . . . . . . . . . . . . . . . . . . . 20

3.2.2.2 3D EBSD technique . . . . . . . . . . . . . . . . . . . . . . 22

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3.2.3 X-ray computed tomography fundamentals . . . . . . . . . . . . . . 24

3.3 Mechanical testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4 Continuum modeling of the mechanical response of a Mg AZ31 alloy

during uniaxial deformation 27

4.1 Experimental procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4.2 Numerical set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4.2.1 Constitutive framework . . . . . . . . . . . . . . . . . . . . . . . . . 28

4.2.1.1 Crystal plasticity continuum formulation . . . . . . . . . . 28

4.2.1.2 Crystal rotation . . . . . . . . . . . . . . . . . . . . . . . . 31

4.2.2 Finite element model . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4.2.2.1 Finite element discretization . . . . . . . . . . . . . . . . . 32

4.2.2.2 Model calibration . . . . . . . . . . . . . . . . . . . . . . . 33

4.3 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4.3.1 Texture evolution and slip/twin activities . . . . . . . . . . . . . . . 38

4.3.2 Fracture mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.3.3 Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.3.3.1 Texture evolution and slip/twin activities . . . . . . . . . . 45

4.3.3.2 Fracture mechanisms . . . . . . . . . . . . . . . . . . . . . 47

4.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

5 3D investigation of grain boundary-twin interactions in a Mg AZ31 alloy

by electron backscatter diffraction and continuum modeling 49


5.2 Numerical set-up and validation . . . . . . . . . . . . . . . . . . . . . . . . . 50

5.3 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

5.3.1 3D variant analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

5.3.2 Effect of GB misorientation on twin transfer . . . . . . . . . . . . . . 59

5.3.2.1 Low misorientation angle boundary (θP1−P2= 15.7°) . . . 60

5.3.2.2 Intermediate misorientation angle boundary (θP1−P3=

22.3°) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

5.3.2.3 High misorientation angle boundary (θP1−P4= 64.3°) . . . 61

5.3.3 Previous model vs. extended model . . . . . . . . . . . . . . . . . . 65

5.3.4 Modeling of twin transfer and twin nucleation as a function of GB

misorientation angle . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

5.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

6 Effect of hydrostatic pressure on the 3D porosity distribution and on

the mechanical behavior of a high pressure die-cast Mg AZ91 alloy 71


6.1.1 Material and process . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

6.1.2 X-ray computed tomography inspection . . . . . . . . . . . . . . . . 73

Contents ix

6.1.3 Image processing and void tracking . . . . . . . . . . . . . . . . . . . 73

6.1.4 Mechanical testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

6.1.5 Finite element analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 75

6.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

6.2.1 Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

6.2.1.1 Qualitative analysis . . . . . . . . . . . . . . . . . . . . . . 76

6.2.1.2 Quantitative analysis of pore volume and shape change . . 79

6.2.2 Simulations results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

6.2.3 Mechanical tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

6.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

6.3.1 Ligament ratio analysis . . . . . . . . . . . . . . . . . . . . . . . . . 88

6.3.2 Effect of the anisotropy on the variation of the volume and the

complexity factor of the pores . . . . . . . . . . . . . . . . . . . . . . 90

6.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

7 Conclusions and future work 95

7.1 Main conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

7.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

A Fractography analysis of a Mg AZ31 alloy 97

B Minimum deviation angle method 99

Bibliography 101

List of Figures 123

List of Tables 129

List of Publications 131

List of Abbreviations

Al Aluminum

AOI Area of interest

ATPs Adjoining twin pairs

AZ31 Mg-3wt.%Al-1wt.%Zn

AZ91 Mg-9wt.%Al-1wt.%Zn

BCC Body centered cubic structure

CF Complexity factor

CPFEM Crystal plasticity finite element method

CRSS Critical resolved shear stress

DTT Double tensile twinning

EBSD Electron backscatter diffraction

FBP Filtered back projection algorithm

FCC Face center cubic structure

FEM Finite element method

FIB Focused ion beam

GB Grain boundary

GNDs Geometrically necessary dislocations

HCP Hexagonal close-packed structure

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HPDC High pressure die-casting

Mg Magnesium

ND Normal direction

ODF Orientation distribution function

OM Optical microscopy

PTR Predominant twin reorientation method

RD Rolling direction

RVE Representative volume element

SEM Scanning electron microscope

SF Schmid factor

STL StereoLithography format

2D Two-dimensional

3D Three-dimensional

3D EBSD Three-dimensional electron backscatter diffraction

TD Transverse direction

VPSC Visco-plastic self-consistent model

XCT X-ray computed tomography

XRD X-ray diffraction

Zn Zinc

Resumen

En los ultimos anos ha habido una fuerte tendencia a disminuir las emisiones de CO2 y

su negativo impacto medioambiental. En la industria del transporte, reducir el peso de

los vehıculos aparece como la mejor opcion para alcanzar este objetivo. Las aleaciones

de Mg constituyen un material con gran potencial para el ahorro de peso. Durante la

ultima decada se han realizado muchos esfuerzos encaminados a entender los mecanismos

de deformacion que gobiernan la plasticidad de estos materiales y ası, las aleaciones de Mg

de colada inyectadas a alta presion y forjadas son todavıa objeto de intensas campanas

de investigacion. Es ahora necesario desarrollar modelos que contemplen la complejidad

inherente de los procesos de deformacion de estos. Esta tesis doctoral constituye un intento

de entender mejor la relacion entre la microestructura y el comportamiento mecanico de

aleaciones de Mg, y dara como resultado modelos de policristales capaces de predecir

propiedades macro- y microscopicas.

La deformacion plastica de las aleaciones de Mg esta gobernada por una combinacion

de mecanismos de deformacion caracterısticos de la estructura cristalina hexagonal, que

incluye el deslizamiento cristalografico en planos basales, prismaticos y piramidales, ası

como el maclado. Las aleaciones de Mg de forja presentan texturas fuertes y por

tanto los mecanismos de deformacion activos dependen de la orientacion de la carga

aplicada. En este trabajo se ha desarrollado un modelo de plasticidad cristalina por

elementos finitos con el objetivo de entender el comportamiento macro- y micromecanico

de la aleacion de Mg laminada AZ31 (Mg-3wt.%Al-1wt.%Zn). Este modelo, que

incorpora el maclado y tiene en cuenta el endurecimiento por deformacion debido

a las interacciones dislocacion-dislocacion, dislocacion-macla y macla-macla, predice

exitosamente las actividades de los distintos mecanismos de deformacion y la evolucion

de la textura con la deformacion. Ademas, se ha llevado a cabo un estudio que

combina difraccion de electrones retrodispersados en tres dimensiones y modelizacion para

investigar el efecto de los lımites de grano en la propagacion del maclado en el mismo

material. Ambos, experimentos y simulaciones, confirman que el angulo de desorientacion

tiene una influencia decisiva en la propagacion del maclado. Se ha observado que los efectos

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no-Schmid, esto es, eventos de deformacion plastica que no cumplen la ley de Schmid

con respecto a la carga aplicada, no tienen lugar en la vecindad de los lımites de baja

desorientacion y se hacen mas frecuentes a medida que la desorientacion aumenta. Esta

investigacion tambien prueba que la morfologıa de las maclas esta altamente influenciada

por su factor de Schmid.

Es conocido que los procesos de colada suelen dar lugar a la formacion de microestruc-

turas con una microporosidad elevada, lo cual afecta negativamente a sus propiedades

mecanicas. La aplicacion de presion hidrostatica despues de la colada puede reducir la

porosidad y mejorar las propiedades aunque es poco conocido su efecto en el tamano y

morfologıa de los poros. En este trabajo se ha utilizado un enfoque mixto experimental-

computacional, basado en tomografıa de rayos X, analisis de imagen y analisis por elemen-

tos finitos, para la determinacion de la distribucion tridimensional (3D) de la porosidad

y de la evolucion de esta con la presion hidrostatica en la aleacion de Mg AZ91 (Mg-

9wt.%Al-1wt.%Zn) colada por inyeccion a alta presion. La distribucion real de los poros

en 3D obtenida por tomografıa se utilizo como input para las simulaciones por elementos

finitos. Los resultados revelan que la aplicacion de presion tiene una influencia significa-

tiva tanto en el cambio de volumen como en el cambio de forma de los poros que han sido

cuantificados con precision. Se ha observado que la reduccion del tamano de estos esta

ıntimamente ligada con su volumen inicial.

En conclusion, el modelo de plasticidad cristalina propuesto en este trabajo describe

con exito los mecanismos intrınsecos de la deformacion de las aleaciones de Mg a escalas

meso- y microscopica. Mas especificamente, es capaz de capturar las activadades del

deslizamiento cristalografico y maclado, sus interacciones, ası como los efectos en la

porosidad derivados de los procesos de colada.

Abstract

The last few years have seen a growing effort to reduce CO2 emissions and their negative

environmental impact. In the transport industry more specifically, vehicle weight reduction

appears as the most straightforward option to achieve this objective. To this end, Mg alloys

constitute a significant weight saving material alternative. Many efforts have been devoted

over the last decade to understand the main mechanisms governing the plasticity of these

materials and, despite being already widely used, high pressure die-casting and wrought

Mg alloys are still the subject of intense research campaigns. Developing models that

can contemplate the complexity inherent to the deformation of Mg alloys is now timely.

This PhD thesis constitutes an attempt to better understand the relationship between the

microstructure and the mechanical behavior of Mg alloys, as it will result in the design of

polycrystalline models that successfully predict macro- and microscopic properties.

Plastic deformation of Mg alloys is driven by a combination of deformation mechanisms

specific to their hexagonal crystal structure, namely, basal, prismatic and pyramidal

dislocation slip as well as twinning. Wrought Mg alloys present strong textures and thus

specific deformation mechanisms are preferentially activated depending on the orientation

of the applied load. In this work a crystal plasticity finite element model has been

developed in order to understand the macro- and micromechanical behavior of a rolled

Mg AZ31 alloy (Mg-3wt.%Al-1wt.%Zn). The model includes twinning and accounts for

slip-slip, slip-twin and twin-twin hardening interactions. Upon calibration and validation

against experiments, the model successfully predicts the activity of the various deformation

mechanisms and the evolution of the texture at different deformation stages. Furthermore,

a combined three-dimensional electron backscatter diffraction and modeling approach has

been adopted to investigate the effect of grain boundaries on twin propagation in the same

material. Both experiments and simulations confirm that the misorientation angle has a

critical influence on twin propagation. Non-Schmid effects, i.e. plastic deformation events

that do not comply with the Schmid law with respect to the applied stress, are absent in

the vicinity of low misorientation boundaries and become more abundant as misorientation

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angle increases. This research also proves that twin morphology is highly influenced by

the Schmid factor.

Finally, casting processes usually lead to the formation of significant amounts of gas and

shrinkage microporosity, which adversely affect the mechanical properties. The application

of hydrostatic pressure after casting can reduce the porosity and improve the properties

but little is known about the effects on the casting’s pores size and morphology. In this

work, an experimental-computational approach based on X-ray computed tomography,

image analysis and finite element analysis is utilized for the determination of the 3D

porosity distribution and its evolution with hydrostatic pressure in a high pressure die-

cast Mg AZ91 alloy (Mg-9wt.%Al-1wt.%Zn). The real 3D pore distribution obtained by

tomography is used as input for the finite element simulations using an isotropic hardening

law. The model is calibrated and validated against experimental stress-strain curves. The

results reveal that the pressure treatment has a significant influence both on the volume

and shape changes of individuals pores, which have been precisely quantified, and which

are found to be related to the initial pore volume.

In conclusion, the crystal plasticity model proposed in this work successfully describes

the intrinsic deformation mechanisms of Mg alloys both at the mesoscale and the

microscale. More specifically, it can capture slip and twin activities, their interactions,

as well as the potential porosity effects arising from casting processes.

CHAPTER 1

Introduction

1.1 Length scales

When a polycrystalline metal is subjected to an external stress, it triggers a heterogeneous

deformation process that is transmitted to each grain. This phenomenon, which involves

changes at all length scales, results in the overall deformation of the material [1]. The

microstructure accommodates plastic deformation through the activation of different

deformation mechanisms: from dislocation movement and interaction causing hardening,

to changes in grain orientation and shape and phase transformations. Numerical

simulations help to better understand the fundamental basis of these deformation

processes. Indeed, computational modeling of materials at the microstructural level

provides a numerical tool able to accelerate the development of modern materials.

Considerable efforts have been made in the development of computational methods to

study the material behavior from continuum length scales (i.e. to predict properties such

as yield stress, elastic modulus, and to understand the evolution of the crystallographic

texture, the particle volume fraction, etc.) to atomic scales (i.e. to predict the atomic

scale processes in a small volume of material for short periods of time, defect energies,

etc.). Figure 1.1 shows the different length scales that are relevant for materials, from the

atomic structure to macroscopic properties.

1.2 Magnesium alloys

Magnesium (Mg) alloys have a density at room temperature equal to 1.74 g/cm3, that is,

one third than that of aluminum (Al) and five times smaller that of iron [2, 3]. For this

reason and for their relatively high specific strength they are excellent candidates to save

structural weight and consequently reduce fuel consumption, especially in the automotive

1

2

Figure 1.1: Multiscale modeling framework of materials across the different length scales.

industry [2–7]. There is also a tendency to promote these materials for aeronautical

applications [8]. These materials have also other useful properties such as shielding against

electromagnetic waves, vibration damping, dent resistance, machinability, and low toxicity

in humans [6]. However, Mg has shortcomings such as insufficient strength, especially at

high temperature, limited elongation and heat resistance, as well as low resistance to

fire and corrosion [6]. Mg alloys are thus still subject to intense research campaigns. In

particular, the micromechanics of deformation are still not fully understood and neither

simulations nor experiments alone can possibly unravel the remaining unknowns.

Mg is an alkaline earth metal which belongs to Group 3 of the periodic table. It

thus has a similar electronic structure to Be, Ca, Sr, Ba and Ra. Indeed, it is the

eighth most abundant element in the earth’s crust and the third most plentiful element

dissolved in the seawater [9]. Mg presents a hexagonal close-packed (HCP) structure. The

lattice parameters for pure Mg estimated at room temperature are a = 0.32092 nm and

c = 0.52105 nm with axes a1 = a2 = a3 = a 6= c and angles α = β = 90°, γ = 120° [9]

(Figure 1.2). The c/a ratio is 1.6236 which is close to the ideal value of 1.633. Therefore,

Mg may be considered almost as perfectly closed packed. The c/a ratio influences the

activation of the different deformation mechanisms in Mg alloys and, in general, in HCP

materials [10].

Various authors have provided overviews on the deformation mechanisms in HCP

metals [11–18]. In any polycrystalline material, five independent systems must be active to

able to undergo a general homogeneous deformation without producing cracks [19]. HCP

metals have a lower number of independent slip systems for each deformation mode than

metals with crystalline lattice body centered cubic structure (BCC) or face center cubic

1.2. Magnesium alloys 3

Figure 1.2: The HCP structure of Mg.

Deformation mode Plane Direction

Basal 〈a〉 {0001} 〈1120〉

Prismatic 〈a〉 {1010} 〈1120〉

Pyramidal 〈a〉 {1011} 〈1120〉

Pyramidal 〈c+ a〉 {1122} 〈1123〉

Tensile twin {1012} 〈1011〉

Compression twin {1011}, {1013} 〈1012〉

Table 1.1: Planes and directions of main deformation mechanisms in Mg and Mg alloys.

structure (FCC). The deformation mechanisms of Mg and its alloys that are operative

at low strain rates have been extensively investigated over the past years [13–18, 20–47].

Plastic deformation is accommodated by slip along the 〈1120〉 direction (〈a〉 direction) in

basal ({0001}), prismatic ({1010}) and pyramidal ({1011}) planes. Moreover, slip along

the 〈1123〉 direction (〈c + a〉 direction) in pyramidal {1122} planes has been observed [36].

Deformation is also accommodated by the activation of twinning, mainly in {1012},

{1011}, and {1013} planes. Twinning is a mechanism which allows plasticity in the

〈1011〉 direction (〈c〉 direction). This deformation mode is a polar mechanism [26], that is,

{1012} twinning is activated by a tensile stress parallel to the c-axis (“tension or tensile

twin”) [40] while {1011} and {1013} twinning occur under compression parallel to the

c-axis (“compression twins”) [41]. At low temperatures and at low strain rates, tensile

twinning is one of the most important deformation mechanisms [15, 30, 33, 36, 43, 48].

Table 1.1 summarizes all slip/twin modes for Mg and Mg alloys (Figure 1.3). The diversity

of possible deformation mechanisms substantially complicates the deformation behavior

of HCP metals in general.

The critical resolved shear stress (CRSS) is the minimum stress required to activate

dislocation movement along a specific slip plane and slip direction. The resolved shear

stress along a specific slip direction on a particular slip plane (σs) must reach the

CRSS for plasticity to occur. This may be expressed as follows for a unidirectional

stress: σs = σncosφ cos λ ≥ CRSS, where σn is the applied stress, and φ and λ are

4

Figure 1.3: Main deformation mechanisms in Mg crystals.

the angle between the stress axis and the slip or twin plane normal and the slip or

twinning direction, respectively. The product cosφ cos λ is termed Schmid factor (SF)

[49]. Although widely spread values have been reported for the CRSS in different slip

and twinning systems for Mg alloys [15, 27, 30, 33, 36, 43], it is generally accepted that

CRSSbasal < CRSStwinning < CRSSprismatic ≤ CRSSpyramidal. Figure 1.4 [50] shows

the variation in the calculated CRSSs for basal and non-basal slip (prismatic slip and

pyramidal slip), as well as for twinning for the Mg AZ31 alloy (Mg-3wt.%Al-1wt.%Zn) at

quasi-static and high strain rates as a function of temperature T . The CRSS for basal slip

and {1012} twinning are believed to be temperature and strain rate independent. However,

the CRSS for prismatic and pyramidal systems decreases with increasing temperatures,

reaching smaller values than the CRSS of {1012} twinning at T > 277 � [9,28,34,50–53].

Twinning takes place profusely at a high strain rate even at temperature as high as

400 � [33, 50,52,53].

The activation of the different deformation modes in Mg and its alloys is also strongly

influenced by the crystallographic texture [10,14,16,22,32,35,37,42,44,50,54–63]. In the

following, the basic understanding of texture is reviewed.

1.3 Texture

Texture is the distribution of crystallographic orientations in a polycrystalline material.

Many materials properties are dependent on texture: Young’s modulus, Poisson’s ratio,

strength, ductility, toughness, magnetic permeability, electrical conductivity and thermal

1.3. Texture 5

Figure 1.4: CRSS variations for the different slip/twin systems as a function of temperatureand strain rate for a Mg AZ31 alloy [50].

expansion [64]. The effect of texture on properties is exploited in materials technology in

order to produce materials with specific characteristics, e.g. in deep drawing processes [65].

The texture has to rely on a quantitative description of orientation characteristics

given by “components” or preferred orientations, and “intensity” which is the volume

fraction of material associated to each component. A weak or random texture is

formed by many components and with very low intensity; a strong texture is formed

by a few components with a very high intensity. In the particular case of metals,

thermomechanical processing leads to the development of specific textures. Fabrication

methods like die-casting processes result in random textures [27, 66], while extrusion,

forging and rolling, promote strong textures and, as a consequence, lead to high anisotropy

[10,44,45,57,58,60,62,66–69].

The most common representation of texture is the so-called direct pole figures,

which are stereographic projections of the orientations of the crystallites. Stereographic

projections allow to transform three-dimensional (3D) information of a crystal into two-

dimensional (2D) representations [64,70]. In order to specify an orientation, it is necessary

to define two coordinate systems, one associated to the specimen and the other to the

crystal. In general, the specimen coordinate system is related to its geometry. For

rolled materials, the axes are: the rolling direction (RD), the direction perpendicular

to the rolling plane (ND) and the transverse direction (TD). Figure 1.5 illustrates how the

(0001) direct pole figure corresponding to a rolled Mg sheet is built. Figure 1.5(a) shows

the dominant orientations of the crystallites. In Figure 1.5(b) one of those crystallites is

placed at the center of a semi-sphere and the (0001) direction is projected on that semi-

sphere. Next, a straight line joining the intersection point and the south pole is drawn.

The intersection between such line and the equatorial plane is termed a (0001) pole. A top-

down view of the equatorial plane and the mentioned (0001) pole is schematically drawn in

Figure 1.5(c). A complete representation of the (0001) direct pole figure requires carrying

out the same procedure for all the crystallites contained in the polycrystal. Representations

6

Figure 1.5: Representation of the direct pole figures of a rolled Mg sheet.

of other (prismatic, pyramidal) direct pole figures involve projecting the corresponding

directions. Figure 1.5(d) illustrates the (0001), (1010) and (1011) poles of a rolled Mg

AZ31 sheet. Alternatively, the orientation of a specific direction in the specimen (i.e. the

extrusion axis in an extruded bar) can be projected onto the crystal coordinate system.

This representation is called an inverse pole figure.

The orientation of any given crystal in space can be obtained from the specimen

coordinate system by performing three rotations given by the so-called Euler angles

(ϕ1,Φ, ϕ2). Although, there are several other conventions for expressing the Euler angles

(Kocks, Roe, Canova) [70], Bunge’s convention is the most commonly used [71, 72]. The

Euler angles are illustrated in Figure 1.6, where XYZ corresponds to the reference specimen

coordinate system and X’Y’Z’ corresponds to the crystallite system. Initially, both systems

are coincident. In a first step, the crystallite system is rotated by an angle ϕ1 around the

Z’-axis (or Z-axis). Then, the crystallite is again rotated by an angle Φ around the X’-axis

1.3. Texture 7

Figure 1.6: Euler angles. The specimen coordinate XYZ is shown in black, the crystallitesystem X’Y’Z’ is shown in red [64].

(in its new position) and, finally, by an angle ϕ2 around the Z’-axis (in its new position). In

general, for a triclinic crystalline structure, the Euler angles are comprised in the following

intervals: 0° ≤ ϕ1 ≤ 360°, 0° ≤ Φ ≤ 180°, and 0° ≤ ϕ2 ≤ 360° [64]. Any given crystal

orientation has an associated orientation matrix g whose elements are functions of the

Euler angles.

g =

(

cosϕ1 cosϕ2 − sinϕ1 sinϕ2 cosΦ sinϕ1 cosϕ2 + cosϕ1 sinϕ2 cosΦ sinϕ2 sinΦ

− cosϕ1 sinϕ2 − sinϕ1 cosϕ2 cosΦ − sinϕ1 sinϕ2 + cosϕ1 cosϕ2 cosΦ cosϕ2 sinΦ

sinϕ1 sinΦ − cosϕ1 sinΦ cosΦ

)

(1.1)

There are two main approaches to measure the crystalline orientation. One involves

averaging a large volume of a polycrystalline aggregate (macrotexture) and the other

focuses on measuring the orientations of individual crystals (microtexture). The

macrotexure is commonly measured experimentally by X-ray diffraction (XRD) or by

neutron diffraction techniques. The direct output from these measurements are incomplete

pole figures in which the information associated to the outer ring is missing due to the

impossibility to obtain sufficient diffracted intensity under low incidence. From these,

the orientation distribution function (ODF), which indicates the probability of finding a

volume element within the material with a specific crystalline orientation, may be obtained.

For the case of HCP metals, five measured pole figures are required to calculate the

ODF. Finally, complete pole figures can be calculated by projecting the 3D information of

the ODF into different planes. Microtexture is measured mainly by electron backscatter

diffraction (EBSD), a technique in which an electron beam is used to scan an area of the

sample surface. At each scanning step the orientation information is stored in the form

of Kikuchi patterns. By comparison of two neighboring orientations, the misorientations

between them may be analyzed. Thus, this method, although lacking the statistical rigor

of macrotexture approaches, provides not only a spatial distribution of orientations but

also a description of the grain boundary (GB) nature (mesotexture).

A GB in a polycrystalline material is a region separating two crystals of the same

phase and different orientation. GBs might be described by means of the corresponding

8

angle/axis pair (θ/r), where θ is the angle (misorientation angle) that one crystal lattice

must be rotated around axis r in order to bring it into coincidence with its neighbor [64].

The misorientation matrix might be calculated from the orientation matrices of two

neighboring grains, g1 for crystal 1 and g2 for crystal 2, by:

M12 = g−11 · g2 (1.2)

From the misorientation matrix M12, the θ/r pair can be calculated as followed:

r1 = m23 −m32

r2 = m31 −m13

r3 = m12 −m21

cos θ = (m11+m22+m33)−12

(1.3)

where mij |i,j=1,...,3 corresponds to the components of the misorientation matrix M12 and,

r1, r2 and r3 are the components of rotation axis r.

For HCP metals, there are 12 crystallographically-related M′

12 matrices which are

generated by premultiplying the misorientation matrix M12 by the symmetry operator

Si |i=1,...,12: M′

12=Si ·M12. Therefore, there are 12 different rotations that could restore

the crystal 2 into coincidence with crystal 1. Each M′

12 is associated to different θ/r pair.

By convention the smallest θ is selected. The GBs can be classified depending on θ [73]:

� Low angle boundary: θ < 5°

� Intermediate angle boundary: 5° ≤ θ < 15°

� High angle boundary: θ > 15°

Rolled sheets and extruded bars of conventional Mg alloys usually have strong textures.

The former consist of a basal fiber with a common 〈0001〉 direction parallel to the normal

direction and the latter of a prismatic fiber with a common 〈1010〉 direction parallel

to the extrusion direction. In Del Valle et al. [67] the distribution of misorientation

angles for a Mg AZ31 alloy with a basal fiber texture, a prismatic fiber texture, and

a random texture were calculated (Figure 1.7). The basal texture is characterized by a

uniform probability between 0° and 30° (the rotation axis coincides with the c-axis). The

prismatic texture is characterized by a uniform distribution in the misorientation angles

between 0° and 90°. Figure 1.7(b) shows the distribution corresponding to a random

aggregate of HCP crystals which has a maximum at 90°, as well as the distribution of

misorientation angles corresponding to a basal fiber with a spread lower than 30° which

presents a maximum at 30°. Therefore a change in texture systematically involves a change

in the GB distribution. The mechanical behavior of Mg alloys is highly dependent on the

character of the misorientation distribution. In fact, the type of GB influences directly

the flow stresses and specific phenomena such as twin propagation [74–76].

1.4. Motivation 9

Figure 1.7: (a) Probability distribution vs. θ corresponding to an aggregate of HCPcrystals with basal and prismatic fiber textures. (b) Probability distribution vs. θ of arandom aggregate and of a basal fiber with a spread lower than 30° (adapted from [67]).

1.4 Motivation

Structural alloys and many emerging multifunctional material systems derive their

desirable combinations of properties largely from the heterogeneity of their structure at

various length scales. The basic deformation mechanisms of a material inherently depend

on the crystal structure, the alloy composition and the processing conditions, which can

cause drastic texture and microstructure changes.

Mg alloys can be classified for structural applications into two important groups

attending to the manufacturing method: wrought Mg alloys (i.e. rolled Mg alloys) and

die-cast Mg alloys. Rolled Mg alloys generally present a strong texture which renders their

deformation strongly anisotropic. This added complexity will lead to the activation of one

deformation mechanism rather than another depending on the direction and orientation

of the applied load. As such, a thorough and comprehensive crystal plasticity model

including both twin and slip systems as well as, more importantly, their interactions

through hardening mechanisms should constitute a powerful numerical tool to relate the

activity of different deformation mechanisms to the mechanical behavior and the texture

evolution. High pressure die-casting (HPDC) is the most common manufacturing process

for Mg alloy components used for automotive as well as for numerous other applications.

Unfortunately, casting processes usually lead to the formation of significant amounts of gas

and shrinkage microporosity, which adversely affect the mechanical properties (ductility,

toughness and fatigue resistance). Consequently, the determination of the 3D micropore

distribution and its incorporation into an adequately calibrated continuum model is of

considerable practical relevance. Systematic research efforts to understand the mechanical

10

behavior of Mg alloys have only been carried out for the last decade, and thus significant

work must still be done before these materials can be widely commercialized [77].

CHAPTER 2

State of the Art

2.1 Rolled Mg alloys: the influence of texture on the

mechanical behavior

Wrought Mg products, such as extruded profiles, rolled sheets and forgings, present high

strength and ductility. However, these alloys also have several limitations. For instance,

the Mg AZ31 alloy which is one of the most common Mg sheet alloys, develops a strong

basal texture and a non-homogeneous recrystallized grain size or partial recrystallization

during the rolling process leading to limited room temperature formability and very high

anisotropy in the mechanical behavior, particularly at room temperature [78].

As mentioned briefly in the previous chapter, slip in HCP metals takes place along the

〈1120〉 (〈a〉) direction on basal and non-basal ({1010}-prismatic, {1011}-pyramidal) planes.

Additionally, second order pyramidal 〈c+ a〉 slip has been observed on {1122} planes [36].

Deformation can be also accommodated by twinning (mainly extension twinning on {1012}

planes). The room temperature CRSS associated with tensile twinning in polycrystalline

Mg alloys, such as AZ31, is moderately higher than the basal slip CRSS, but significantly

lower than that of any non-basal slip CRSS [15, 27, 30, 33, 36, 43]. In agreement with this

trend, earlier studies reported that slip on basal planes and {1012} twinning are the main

deformation mechanisms during uniaxial deformation at low temperatures and low strain

rates in randomly oriented Mg polycrystals of conventional grain sizes (∼ 10 − 50 µm)

[13–15,17,20,22]. Non-basal slip systems are also active, albeit to a lesser extent [32,36,38].

The activity of different deformation modes is highly dependent on temperature and initial

texture [35,44].

Rolled Mg sheets usually present a strong basal texture, i.e. the c-axes of most grains

are parallel to the ND. In this case, when compression is carried out along ND, both

basal slip and {1012} twinning are not favored and thus non-basal slip [36] and {1011}

11

12

twinning [38] are activated at the early states of deformation. Non-basal slip systems are

also activated when tension tests are carried out along RD or TD [36,38]. On the contrary,

when a Mg sheet is compressed at room temperature along RD, i.e. the compression axis

is perpendicular to the c-axes in most grains, {1012} twinning predominates at the early

stages of deformation, causing grains to rotate ∼ 86.3° [26] and thus the c-axes to become

closely aligned to the ND. Significant strain hardening ensues due to slip-slip, slip-twin

and twin-twin interactions. The activation of different deformation mechanisms due to

the different angles between the loading direction and the c-axis causes an asymmetry in

the yield stress [25,34,43].

Non-basal dislocations, including 〈c+ a〉, have often been reported as a necessary

accommodation for deformation in regimes of high stress concentration [32, 79, 80].

However, the net contribution of second-order pyramidal 〈c+ a〉 slip to the macroscopic

strain in Mg alloys has been controversial for decades. This mechanism, if active, would

provide six independent deformation modes, thus increasing significantly the ductility

and the isotropy of Mg alloys. The activity of 〈c+ a〉 slip during room temperature

deformation is still a subject of debate. Due to its high CRSS, this deformation mode

may only be activated when the compression axis is parallel to the 〈0001〉 axis, as neither

the basal, the prismatic nor the tensile twinning systems are favored. This occurs, for

example, in the following two cases. The first case is a rolled and annealed Mg sheet

with strong 〈0001〉 texture subjected to compression along ND. Some authors have indeed

reported the occurrence of 〈c+ a〉 slip under such conditions [16, 27], although others

have found evidence of the accommodation of such deformation by a double twinning

mechanism [55, 81]. The second case is when, during compression of the same strongly

texture sheet along RD, twinning orients most grains with the c-axes parallel to the

compression axis during the first stages of deformation (i.e. extension twinning results

in an 86.3° reorientation of the basal pole approximately [26]). The operation of 〈c+ a〉

slip during the last stages of deformation has been reported under such conditions by

[59]. Other investigations report that strain is, instead, accommodated by the activation

of other mechanisms, such as a combination of basal and pyramidal 〈a〉 slips [3], the

increasing activity of basal slip [43] or the simultaneous operation of basal slip and double

twinning [81].

Such controversy is by definition difficult to solve by experimental means, requiring

in situ simultaneous tracking of slip/twin system activities and texture evolution. In this

example as in many others, it is thus convenient to approach the problem with the help

of crystal plasticity numerical simulations (Section 2.4).

2.2 Fundamentals of twinning

Twinning is a key deformation mechanism in HCP metals [26] and, in particular, in Mg

alloys [78]. The propensity for twinning has a tremendous impact on the mechanical

behavior of these materials, as it dramatically affects their yielding behavior [60], work

2.2. Fundamentals of twinning 13

hardening and ductility [51, 63, 82], as well as the activation of the different dynamic

recrystallization mechanisms [44, 83–86]. The twinning activity in Mg alloys is highly

dependent on the testing conditions and is, in general, enhanced with increasing strain

rate and low temperatures [52,87].

Mg alloys twin mainly on {1012}, {1011} and {1013} planes [26,78]. The {1012}〈1011〉

system, consisting of six equivalent variants, gives rise to a shear (γ = 0.1289) that results

in an extension of the c-axis [40]. The room temperature CRSS associated with tensile

twinning in pure polycrystalline Mg and in conventional alloys such as Mg AZ31 alloy is

moderately higher than the basal slip CRSS, but significantly lower than any non-basal slip

CRSS. Thus, profuse tensile twin nucleation, propagation and growth are observed at this

temperature. Twinning on {1011} and {1013} planes is associated with a compression

along c-axis [41]. These compression twins are much less abundant, because the room

temperature CRSS corresponding to this deformation mode is relatively high, of the order

of the non-basal slip CRSS [87], and they involve larger shuffles than tensile twinning [26].

Thus, the latter has been more profusely studied. In this work, the term twinning will

hereafter refer to tensile twinning. The influence of microstructure on twin nucleation,

propagation and growth phenomena is complex and still not well understood [34,61,88,89].

Grain orientation has been found to strongly influence the likelihood of twin nucleation

and, to a greater extent, of twin growth. Accordingly, grains favorably oriented for

twinning have a larger probability of forming at least one twin, and variants with the

highest SF tend to grow to a greater length [34, 61, 88–90]. However, twinning has been

observed in grains with a wide range of SF [89–94]. Variant selection has been linked to the

accommodation work needed in neighboring grains [93,94] in such a way that twins with

low SF may be preferred if the associated accommodation strains result in a better overall

energy balance. In general, the formation of secondary and tertiary twin variants has been

related to the presence of local stresses that may differ significantly from the externally

applied stress [48,92]. The effect of grain size on twinning has been intensively investigated.

It has been commonly reported that twinning is enhanced with increasing average grain

size [30, 34]. However, recent statistical studies [48, 89] have revealed that, while the

number of twins per grain is strongly related to grain size, the probability of twinning is

independent of the grain diameter, at least when the latter is larger than a few microns,

thus questioning the validity of a Hall-Petch dependence on twin nucleation [34,58,95,96].

The influence of the nature of GB on twinning has been significantly less studied.

GB are favorable sites for twin nucleation, as this process requires large stresses and

large defect sources [48, 74]. Quantifying the influence of GB misorientation on twinning

is a challenging task. Common post-mortem characterization techniques such as EBSD

do not allow one to distinguish whether, for example, two twins that meet at a GB,

termed adjoining twin pairs (ATPs), are a result of nucleation of the twin pair or of

twin transfer across the boundary. Recently, Beyerlein et al. [48, 89] and El-Kadiri

et al. [75, 76] have shown clear evidence that ATPs occur most frequently at low to

moderate misorientation GB (θ < 15°), suggesting that these GBs constitute preferred

14

sites for twin nucleation. These studies further ascribe the non-Schmid behavior of

twinning to local stress fluctuations at GB. Similar effects have been also considered

in FCC metals [97, 98]. However, the influence of GB on twinning is still not well

known. In particular, further studies are required to explain the influence of specific

GB misorientation angles on plasticity in the vicinity of the boundaries as well as how the

latter influences macroplasticity of Mg alloys.

2.3 Die-cast Mg alloys: the influence of porosity on the

mechanical behavior

The vast majority of all Mg components are produced by die-casting. They have several

advantages such as lightness, high specific strength, good machinability, good castability,

low melting temperature and melting energy [3, 7]. In particular, HPDC is the leading

processing technology for Mg components in the automotive industry as it allows forming

parts with complex geometries in one single operation with a limited cost.

In HPDC the molten metal is injected into the die at a high speed until the cavity is

completely filled. In order to reduce the volume fraction of pores and casting defects, a

pressure of approximately 400-500 bar is subsequently applied and withdrawn only when

the solidification process is complete. The heat released by the part is mostly eliminated

through the casting-die interface [99]. Finally, the part is rapidly removed from the die.

The most common die-cast Mg alloys belong to the Mg-Al series (AM60, Mg-6wt.%Al-

0.5wt.%Mn and AZ91, Mg-9wt.%Al-1wt.%Zn).

Even though HPDC was invented more than a hundred years ago, relatively few studies

relating the casting parameters to the resulting microstructures exist to date [100–103].

This is motivated by the large cost of setting-up and maintaining HPDC facilities in a

research laboratory. A large number of studies have tried to relate HPDC microstructures

to their mechanical behavior [104–116]. In general, it is agreed that porosity is the

microstructural feature that has the most deleterious effect on the mechanical properties.

In particular, the presence of pores gives rise to fracture at smaller strains than in

their wrought counterparts. Furthermore, the current impossibility to reproduce 3D

pore distributions in different HPDC cycles leads to a large variability in the tensile

ductility of different components, thus constituting a serious limiting factor for the wide

commercialization of HPDC Mg parts. Fracture has been reported to initiate at the largest

microvoids [113] and at porosity segregation zones [111]. It has been proposed that tensile

ductility is not related to the bulk volume fraction of pores, but to the area fraction of

the pores at the fracture surface [107–110,114,115].

It is known that porosity and its negative effects on ductility might be reduced by

applying a hydrostatic pressure [117, 118]. Biner et al. [117] evaluated the effects of

hydrostatic pressures up to 1,104 MPa on the densification of iron compacts with porosity

levels ranging from 0.3% to 11.1%. The decrease in porosity resulting from pressurization

2.4. Continuum models for HCP materials 15

was more pronounced in samples with higher initial volume fraction of pores. The variation

of the porosity level with the applied hydrostatic pressure was found to agree well with

a modified Gurson’s model including internal pressure in gas pores [119, 120]. Uniaxial

deformation under pressure has also been found to result in higher tensile ductilities due to

the progressive suppression of macrovoid formation within necked regions [118, 121, 122].

At sufficiently high pressures, the fracture mode was observed to change into a shear

dominated mechanism.

Quantification of the 3D porosity distribution in die-cast alloys is a very complex task.

However, this would significantly help to improve casting processes in order to design

cast components with improved properties. Moreover, inclusion in materials models of

a realistic description of the porosity distribution would undoubtedly constitute a major

step forward towards the design of advanced die-cast Mg alloys.

2.4 Continuum models for HCP materials

The mechanical behavior of polycrystalline materials, such as Mg alloys, can be

investigated by using different experimental techniques, which allow to characterize

and analyze several microstructural features and provide fundamental information and

understanding. However, experimental approaches usually require the use of sophisticated

equipment, careful material manufacturing and preparation and complicated post-

processing. They are also generally expensive and time consuming. Computational

mechanics thus appears as a valuable tool to complement experimental campaigns.

Two modeling approaches aimed at tackling the difficulties characteristic of rolled (i.e.

strong anisotropy) and die-cast (i.e. porosity evolution) Mg alloys discussed previously,

are presented in this section.

2.4.1 Crystal plasticity modeling

Despite the availability of other simulation techniques, such as dislocation dynamics

[123–127] or atomistic simulations [125, 128, 129], the finite element method (FEM)

remains by far one of the few techniques able to accurately describe polycrystalline plastic

deformation while avoiding the drastic length and time scale limitations plaguing the

two other techniques. Note that there has been some attempts to bridge scales between

techniques [130,131], but without fully alleviating these restrictions. The crystal plasticity

finite element method (CPFEM) has increasingly gained momentum in the field of metals

modeling and particularly in multiscale mechanical and micromechanical modeling. In

these approaches one typically assumes the stress response at each macroscopic continuum

material point is given by one crystal or by a volume averaged response of a set of

grains. The latter method naturally involves local homogenization [132]. Homogenization

techniques [27, 133–143] allow by definition for homogenized calculations of an overall

16

polycrystal with a decreased computational effort, but are consequently not as flexible as

FEM for microstructural evolution studies (e.g. Zhao et al. [144]).

If FCC and BCC crystal plasticity continuum models have already been extensively

studied [132, 138, 145–157], similar efforts have been much scarcer for HCP metals

[62, 158–165]. Despite the relatively ancient interests in HCP metals, the intrinsic

mechanisms of HCP crystalline deformation, such as twinning (generally more present than

in FCC and BCC metals), twin-slip interaction, or twin polarization among others, are

still, as of now, not fully understood [128,129,150,166]. Among these efforts, Staroselsky

and Anand [158,159] developed a non-hardening rate-independent model of Mg AZ31 alloy

aimed at reproducing the experimental behavior of tension and compression tests. Graff

et al. [160] presented a crystallographic model of Mg but without taking into account the

reorientation of the twinned volume. Mayeur and McDowell [161] developed an HCP model

for Ti-6Al-4V alloy, in which twinning is less likely to occur. Tang et al. [162] proposed a

crystal plasticity model aimed at simulating cup drawing of Mg AZ31 alloy and Mayama

et al. [62] studied the effect of twinning deformation and rotation in a crystal plasticity

model on pure Mg deformation. Prakash et al. [167] modeled the evolution of texture of Mg

AZ31 alloy but without differentiating between the different cross-hardening mechanisms.

Choi and coworkers [163, 164] proposed a similar model focusing their analysis on stress

concentration and texture evolution of Mg AZ31 alloy at high temperature, in which case

the sequence of mechanisms of deformation is intrinsically different. Levesque et al. [165]

developed a model for Mg AM30 alloy (Mg-3wt.%Al-0.5wt.%Mn) at high temperature

but without taking into account slip-twin interaction. Finally, Izadbakhsh et al. [168]

proposed a complex model a priori involving primary, secondary and tertiary slip systems

with primary and secondary contraction/extension twin systems. Because of the significant

amount of accounted mechanisms, this approach however involves a daunting number of

parameters difficulty identifiable experimentally, and leading to the non-uniqueness of the

solutions.

2.4.2 Mechanical twinning in CPFE models

Early material models consider slip as the major plastic deformation mechanisms (Taylor

[19], Peirce et al. [169,170] and Asaro and Needleman [171]). It is well known that plastic

deformation due to crystallographic slip induces a gradual texture evolution as a function of

imposed strain. However, tensile twinning in Mg alloys results in an abrupt 86.3° rotation

in the crystal lattice of a grain [26]. Therefore, special considerations are needed for the

twinning mechanisms and hence, their incorporation into suitable continuum models.

Incorporating twinning into models of plasticity in Mg alloys has proved challenging.

Two main approaches have been proposed [132]: one typically assumes the stress response

at each macroscopic continuum material point to be potentially given by one crystal, and

the other considers a volume averaged response of a set of grains comprising the respective

material. The latter is named homogenization schemes such as visco-plastic self-consistent

2.4. Continuum models for HCP materials 17

(VPSC) models [27, 133, 136, 139–141, 172]. Both approaches can successfully predict

strain hardening and texture evolution during plastic deformation [27, 159]. However,

in the attempt to capture twinning phenomena, some differences arise. Van Houtte [173]

proposed a model where the swift crystallographic rotation due to twinning is driven by a

statistical criterion based on the volume fraction of the twinned regions in the grain and

entire polycrystal aggregate. This assumption is called predominant twin reorientation

(PTR) method and twinning is considered to be a pseudo-slip mechanism [83], and a grain

is allowed to reorient if an accumulated value reaches a specified threshold. The PTR

scheme has been successfully implemented in VPSC models [172, 174, 175], as well as in

CPFEM [158, 159]. The effect of deformation twinning promotes a heterogeneous stress

concentration inside the grains as well as in the GBs. Some efforts have been carried out

to study the twin-parent (original grain) grain interaction: one investigates the stress state

inside the twin at twin inception [176] and another load sharing or interaction between

child (new twinned domain) and parent [175]. These efforts are based on self-consistent

models which typically assume a uniform strain state within a grain and as a consequence,

several effects like misorientation between two neighboring grains and grain diameter or

volume on nucleation of twins or twin variant selection, as well as, stress concentration due

to GB geometry are ignored. Recently, Beyerlein et al. [48, 74] have proposed to include

a probabilistic approach that takes into account fluctuating stresses at GB. Alternatively,

models based on CPFEM can potentially successfully predict the heterogeneous stress

concentration at the grain level during deformation as well as the activation of more than

one twin variant in the parent grain. Despite some efforts in this direction [177–179], the

existing CPFE models to date are largely insufficient to explain how deformation twinning

affects the stress concentration at GBs or twin boundaries in Mg AZ31 alloys during plastic

deformation.

2.4.3 Continuum models for porosity evolution

Die-cast Mg alloys present in general a certain volume fraction of microporosity which

adversely affects the mechanical behavior. Additionally, the microstructure is composed

by dendrites and irregularly shapes of crystals which are affected by the inherent anisotropy

of HCP structures. As a consequence, modeling the mechanical response of these alloys

remains a major challenge.

Two main approaches can be considered: dilatational plasticity and polycrystal

plasticity. Early models of dilatational plasticity based on limit analysis of an isotropic

hollow sphere were proposed by Rice and Tracey [180] and developed later by Gurson [120].

A study based on the combination of dilatational plasticity and single crystal plasticity

was carried out to study void growth in single crystals [181]. This study showed a strong

effect of the intrinsic crystal plastic anisotropy on void growth. Moreover, homogenization

formulations based on classic self-consistent models have been proposed to capture some

effects related to the polycrystalline character of the matrix in the voided aggregate [182].

18

A recent study based on the fast Fourier transform [183] has been proposed along

those lines [184]. This sophisticated method allows to compute void growth in porous

polycrystalline materials, deforming in a dilatational viscoplastic regime. However, this

formulation requires periodic microstructures and regular computational grids, making

them less general than classical CPFE models.

In the particular case of casting materials, several studies have aimed, modeling

the correlation between ductility and porosity [104–106, 116]. Some have attempted to

include 3D characteristics of the porosity distribution into models of cast alloys [185–187].

The 3D geometry of the pores was measured using X-ray computed tomography (XCT)

techniques or by serial sectioning combined with metallographic observations. In Ref. [185]

a significant fraction of pores is measured and the methodology to include this fraction in

a FE mesh is described. In Ref. [186] a single pore is taken into account. Vanderesse et

al. [187] utilize a relatively large microstructural volume of a cast Al alloy as an input for

FE calculations. Further work is clearly needed in this area if realistic models of HPDC

Mg alloys with good predictive capability are to be developed.

2.5 Objectives

This PhD thesis constitutes an attempt to improve toward the design of advanced Mg

alloys and it will result in the definition of polycrystalline models integrating both the

texture and the porosity distribution, which will allow for the prediction of the mechanical

behavior of these materials under a wide range of conditions.

The project is divided into several specific objectives:

1. Development of a CPFE model for HCP materials deforming by slip and twinning

(and where both can interact) and application of this model to a rolled Mg AZ31 alloy

sheet. In this initial stage, a simplified cubic grain topology will be used.

2. This CPFE model will then be used to represent more realistic polycrystalline

features accounting for the topology of the grains. In the extended model, a 3D polycrystal

will be represented as a 3D Voronoı tessellation with multiple elements per grain, thus

allowing for the study of the local intragranular mechanical fields. The validation will be

carried out with a rolled Mg AZ31 alloy sheet.

3. Analysis of twin-GB interactions in a rolled Mg AZ31 alloy sheet by three-

dimensional electron backscatter diffraction (3D EBSD) and using the extended CPFE

model previously developed.

4. Study of the effect of hydrostatic pressure on the porosity distribution in a die-cast

Mg AZ91 alloy by XCT. Development of a FE model making use of an isotropic hardening

law as a first approximation. The model will represent a real 3D pore distribution of a

die-cast Mg AZ91 alloy obtained from XCT. Study of the effect of anisotropy of the HCP

lattice of Mg alloys on the evolution of the porosity with the hydrostatic pressure.

CHAPTER 3

Experimental Campaign

3.1 Material

The materials investigated are commercial Mg alloys with additions of Al and Zn (AZ

series). Al is the most commonly used Mg alloying element as it is one of the few metals

that dissolves easily in Mg, improving both strength and hardness. Zn is the second

most commonly used alloying element and contributes to improve the strength at room

temperature as well as the corrosion resistance. Moreover, AZ-based Mg alloys contain

manganese, which provides high ductility and fracture toughness as well as better corrosion

resistance [3,4]. In particular, two Mg alloys are used in this thesis: AZ31 and AZ91 alloys.

3.2 Microstructural characterization

The microstructure of the investigated materials has been analyzed by optical microscopy

(OM). The details of the apparatus utilized are included in the corresponding following

chapters. The macro- and microtextures have been examined by XRD and EBSD.

Furthermore, this thesis constitutes an attempt to generate 3D representations of

the microstructure by the use of novel techniques such as 3D EBSD and XCT.

3D characterization of a sample is critical in order to have a correct understanding

of the role microstructure plays in a particular material property. Moreover, 3D

observations are of considerable practical relevance for computational modeling where

realistic microstructural information is of vital importance to render accurate simulations.

Detailed information about these techniques is provided below.

19

20

3.2.1 Measurement of macrotexture by X-ray diffraction

Macrotextures were measured by the “Schulz reflection method” [64, 188]. Figure 3.1(a)

shows a picture of the X-ray diffractometer utilized. This equipment is furnished with

a texture goniometer which is depicted schematically in Figure 3.1(b). A sample with

a flat surface is placed in a special holder located at the inner surface of the Euler ring

(goniometer) in such a way that the irradiated surface always contains a tangent to the

focus circle intersecting the X-ray source and the detector. The ring rotates around its

vertical diameter (angle ω) until the X-ray incidence angle corresponds to the desired

Bragg reflection (h=(h,k,l)) [64, 188]. Then, the sample holder is rotated at steps of

3° around a direction perpendicular to the ring (angle χ) in order to allow the (h,k,l)

planes of the different crystallites to reach the corresponding exact Bragg position. At

each χ step, the sample rotates 360° around an axis perpendicular to the sample surface

(angle ϕ). At each sample position, the diffracted intensity is collected by the detector.

χ varies from 0° to 75°, as higher angles give rise to a very small diffracted intensity.

The information collected using this method is later utilized to calculate the ODF from

which, subsequently, the calculated pole figures are derived, as explained in Section 1.3.

The specific characteristics of the diffractometer and the radiation utilized, as well as the

sample preparation procedure, are described in the corresponding following chapters.

(a) (b)

Figure 3.1: (a) X-ray diffractometer used in this work. (b) Schematic illustrating thetexture goniometer and the corresponding rotation angles.

3.2.2 Measurement of microtexture by electron backscatter diffraction

3.2.2.1 2D EBSD technique

Microtextures were measured by EBSD. Figure 3.2 shows a schematic of the EBSD system.

It consists basically of a scanning electron microscope (SEM), a CCD camera and a

computer with the software to analyze the orientation data. The samples are placed

into the SEM in a special holder with a high inclination (usually 70°) to the horizontal.

This reduces the penetration of the beam and therefore, the absorption, and ensures that

3.2. Microstructural characterization 21

Figure 3.2: Schematic drawing of the EBSD system.

the diffracted signal is more intense. The inclination of the holder has been optimized,

moreover, to guide the diffracted signal towards the CCD the camera.

The electron beam hits an individual crystallite and is diffusely scattered in all

directions. There must always be some electrons arriving at the Bragg angle (θB) at

every set of lattice planes, and these electrons can then undergo elastic scattering to give

a strong, reinforced beam. Since diffraction of the electrons through θB is occurring in all

directions, the diffracted radiation emerges along the surface of a cone (Kossel cone) [64].

These Kossel cones will intersect the phosphor screen, forming the so-called Kikuchi lines,

which are shown in Figure 3.3(a). Each pair of parallel lines represents a lattice plane and

the width between lines is an angular distance of 2θB which in turn is proportional to the

interplanar spacing. The set of Kikuchi lines corresponding to an individual crystallite

forms a Kikuchi pattern (Figure 3.3(b)).

(a) (b)

Figure 3.3: (a) Origin of Kikuchi lines [64]. (b) Kikuchi pattern.

22

The image of Kikuchi patterns on the phosphor screen is captured by the camera. After

being integrated and filtered, the signal is digitized and sent to a work station provided

with a program that helps to determine the orientation and save the data analyzed in a file.

As the beam sweeps the sample along a predefined path, orientation data from adjacent

positions will be acquired and recorded. When comparing two consecutive orientations it is

possible to obtain information about the misorientation or the nature of the grain boundary

that separates them. Then the software is employed to produce orientation image mapping,

pole figures representing the texture, disorientation distribution histograms, etc.

3.2.2.2 3D EBSD technique

Fully automated 3D orientation microscopy systems combining SEM, focused ion beam

(FIB) and EBSD have been developed in the last few years [64,189–191]. FIB uses a finely

focused beam of gallium ions (usually 30 kV Ga+) to mill slices of the target material.

After each slice has been removed, the EBSD technique is used to obtain a 2D orientation

map from the fresh surface (Figure 3.4). This cycle is repeated several times. Finally,

all the 2D maps are reassembled and the volume of interest is reconstructed. This serial

sectioning method is extremely laborious. The main challenge is controlling the sectioning

depth, obtaining flat and parallel surfaces, and correctly redetecting and aligning the 2D

maps corresponding to each section.

Figure 3.4: Schematic representation of the grazing-incidence edge-milling method [192].

The 3D EBSD technique used in this work was developed by the Max-Planck Institute

for Iron Research in Dusseldorf, Germany (MPIE) in collaboration with EDAX/TSL and

Zeiss, Figure 3.5(a). The FIB column is positioned opposite to the EBSD camera [192]

(Figure 3.5(b)), which allows a precise and quick change of the sample from the FIB

position (stage tilt of 34°) to the EBSD position (stage tilt of 0° and sample tilt of 70°). The

set-up for 3D EBSD measurements corresponds to that followed by A. Khorashadizadeh

[193]. The sample is fixed on a holder and it is inserted into the microscope. First, the

sample is aligned for EBSD analysis and then the stage is tilted into the FIB position (34°).

After identifying the SEM and FIB beam cross-over point, the FIB position is saved. In


(a)

(b)

Figure 3.5: (a) (FIB)-FEG-SEM dual beam instrument; (b) Schematic representation ofthe tilt geometry [192].

order to find the precise position after every new cycle, a cross marker is performed by

milling on the sample surface close to the current measurement area. This cross is detected

afterwards at the beginning of each new milling process. In order to prevent shadowing

from the side walls of the analyzed area during the process, these side walls are milled with

a current of 2 nA, and thus, the investigated area is bounded by 55° of side walls. The fine

milling is performed using a beam current of 500 pA. The volume of material that can be

analyzed depends on several circumstantial factors: amount of instrument time available,

the fidelity of the data required, the nature of the studied problem and the amount of

effort required. However, the principal limiting factor is given by the edges of the pattern

which can become shadowed when a given width is reached.

24

The milling method used in this work is named grazing-incidence edge-milling

[190, 192]. For this method, the sample is required to have a sharp rectangular edge,

which can be obtained by thorough mechanical grinding and polishing. Therefore, milling

is performed at grazing-incidence to one surface of this sample edge. Tremendous efforts

have to be done in order to obtain an area of the interest of the microstructure close to

the sample edge. The schematic strategy is represented in Figure 3.4.

Reassembly of all the 2D EBSD maps to reconstruct a 3D orientation volume is carried

out using the Qube software developed by MPIE [194]. Initial preparation of the individual

slice images is performed using the batch processing function of the EDAX-TSL OIM

Analysis software package. This function allows cleaning-up all the 2D images, map

construction and image storage. Subsequently, all 2D EBSD maps will be ready to be

imported into the 3D software.

3D EBSD has proven to be a powerful technique for 3D microstructure characterization

on several issues such as recrystallization [85,195], microstructure and texture investigation

of nanostructured materials [196], plastic zones below nanoindents in single crystals

[197–199], and fatigue cracking [192]. In this work 3D EBSD will be utilized to obtain a

better understanding of the twinning mechanism in Mg alloys.

3.2.3 X-ray computed tomography fundamentals

XCT is a non-destructive technique for the characterization of material microstructure in

3D at micron level spatial resolution [200–202]. It shows all the microstructural features

such as phases, inclusions, cracks, pores, etc., provided the spatial resolution is high

enough. These features are obtained through the modification of the attenuation (in the

case of absorption tomography) or of the optical phase (phase contrast tomography) along

the path of X-rays. The 3D information provided by XCT can be extracted and quantified

by applying image analysis procedures to the reconstructed volumes. These characteristics

make XCT a powerful technique that has been used over the past twenty years in material

science research for microstructural and defects characterization damage assessment,

dimensional inspection, non-destructive testing, to name few [203, 204]. Additionally a

major advantage is that no careful specimen preparation is needed.

XCT is based on X-ray radiography, which represents a “projection” of the X-

ray absorption coefficient along the X-ray path through the investigated material. A

tomography scan or tomogram consists in recording X-ray radiographies at many different

angles, i.e. by rotating the object about a single axis. Therefore, the tomogram is

mathematically reconstructed to produce a 3D map of X-ray absorption in the volume.

The 3D digital volume is typically presented as a series of 2D slices images where each

voxel (i.e. a volumetric pixel) represents the X-ray absorption at that point. A typical

tomographic set-up is observed in Figure 3.6, in which the object is located between the

X-ray source and the detector. The X-rays, emitted from the source, pass through different

parts of the specimen placed on a rotation stage and the signal is attenuated by scattering


and absorption. There are three dominant physical processes responsible for attenuation

of an X-ray signal: photoelectric absorption, Compton scattering and coherent scattering.

The photoelectric effect is the dominant attenuation mechanism at low X-ray energies, up

to approximately 60-100 keV. Compton scatter becomes important at higher energies up

to 5-10 MeV. Thus, for X-ray absorption tomography the energy range considered varies

between 6 to 100 keV.

Figure 3.6: Schematic of a XCT acquisition configuration for cone beam geometry [205].

The X-ray intensity I(x, y) is acquired by a photo detector which records the

radiographies. Mathematically, a radiographic projection can be described according to

Lambert-Beer’s law [206, 207]. It relates the ratio of the transmitted (I) to the incident

(I0) intensity to the integral of the linear absorption coefficient of the material (µ), which

depends on the material and on the X-ray energy, along a straight path s(x, y). For a

monocromatic and parallel X-ray beam it can be written as:

p(t) |θ = − lnI

I0=

∫

Γµ(x, y)ds (3.1)

In the (x,y) plane (normal to the axis of rotation) the object can be represented by a

spatial function of X-ray absorption, f(x, y). Therefore, an absorption profile p(t, θ) on an

axis t is created by the projection of any set of parallel X-rays passing through the object.

The intensity profile is acquired for each θ increment (Figure 3.7).

Once the projections are recorded, the next step is to obtain the tomographic

reconstruction itself which is based on Radon transform [206, 207]. This states that

the reconstruction of the object f(x, y) with unknown density is possible from the X-

ray attenuation projections acquired at infinite rotation angles, p(t, θ). Then, the Radon

transform represents the scattering data obtained as the output of a tomographic scan.

The theory governing the tomographic reconstructions is given by the Fourier slice theorem

26

Figure 3.7: Schematic of the X-ray absorption model used for tomographic reconstruction[201].

(or projections theorem) which gives a connection between the Radon and the Fourier

transform. In the case of parallel beam geometry, the slices of the sample corresponding

to different heights can be treated independently and it is also sufficient to record the

projections for half turn due to mirror symmetry, i.e. p(t, θ + π)=p(−t, θ). The Fourier

transform provides a direct solution to the reconstruction problem, but its implementation

is hard and unstable. In practice, the filtered back projection (FBP) algorithm is one of

the most common methods for tomographic reconstruction [206,207]. FBP states that the

back projection is obtained by assigning to each point of the object the average intensity of

all the projections that pass through that point. The back projected image is, however, a

blurred version of the original object. Therefore, the reconstructed object is filtered using

a high pass filter in combination with a window to decrease noise at high frequencies.

The XCT technique has been used in this work to analyzed the 3D porosity distribution

of an HPDC Mg AZ91 alloy. In particular, the morphology of cavities, the cavity spatial

orientation, and the minimum distance between cavities were characterized. Measurements

were carried out in Nanotom 160NF tomograph (Phoenix, Inc.). The specific working

conditions are described in detail in Chapter 6.

3.3 Mechanical testing

Uniaxial and plane strain tests have been carried out at room temperature and quasi-static

strain rates in order to investigate the mechanical behavior of the materials under study.

Tests were performed in both an electromechanical Servosis ME 405/10 testing machine

and an Instron 3384 system. The specific testing conditions as well as the geometry of the

coupons are described in detail in the corresponding following chapters.

CHAPTER 4

Continuum modeling of the mechanical response of a Mg AZ31 alloy

during uniaxial deformation

In this chapter a CPFE model is developed to predict the mechanical behavior of rolled Mg

alloys. This model complements the model of Staroselsky and Anand [158,159] by adding

differentiated self- and cross-hardening between slip and twin systems and focusing on the

individual calibration of each one of the interaction parameters. The model is calibrated

and validated against uniaxial compression tests of Mg AZ31 alloy along both RD and

ND. The evolution of the texture, the slip/twin activity, and the stress-strain response are

analyzed and compared against experimental results for both cases. Finally, the model is

used to cast light on the controversy on the role of pyramidal slip and a failure criterion

based on slip activity is proposed.

4.1 Experimental procedure

The starting material is a sheet of the Mg AZ31 alloy, with a thickness of 3 mm and a

grain size of ∼ 13 µm. The composition of this material is summarized in Table 4.1.

Uniaxial compression tests were carried out at room temperature and at 10−4 s−1 along

RD in an electromechanical Servosis ME 405/10 testing machine. The specimens were

3 mm×3 mm×4.5 mm prisms. With the aim of analyzing the evolution of texture during

compression, tests were interrupted at several intermediate strains: 2%, 4%, 6%, 11%, and

Al Zn Mn Ca Si

2.5-3.5 0.7-1.3 0.20 min 0.04 max 0.30 max

Cu Ni Fe Others Mg

0.05 max 0.005 max 0.005 max 0.030 max remaining

Table 4.1: Composition of the Mg AZ31 alloy under study (in wt. %).

27

28

14%. Additional compression tests of 3 mm×3 mm×3 mm specimens were carried out

along ND under the same conditions of temperature and strain rate. A note should be

made that compression along RD is equivalent to a compression perpendicular to the c-

axis while compression along ND is equivalent to a compression parallel to the c-axis. For

both compression cases (RD and ND), three tests were carried out exhibiting a reasonable

reproducibility, and the two experimental stress-strain curves used for calibration (Figure

4.4) correspond in both conditions to the average of the three associated tests. The textures

of both the as-received and deformed Mg AZ31 alloy samples were measured by the Schulz

reflection method in a Philips X’Pert-Pro Panalytical X-ray diffractometer furnished with

a PW3050/60 goniometer. The radiation used was β-filtered Cu Kα. The surface area

examined was about 2 mm2. The polar angle ranged from 0° to 75° in steps of 3°. The

irradiation time at each step was 2 s. The measured incomplete (1010), (0002), (1011),

(1012), (1120) and (1013) pole figures were corrected for background and defocusing using

the Philips X’Pert software. The MTEX Matlab open source software [208] was then

utilized to calculate the ODF from the corrected pole figures and, from the ODF, the final

complete pole figures. Sample preparation for texture measurement included grinding

with increasingly finer SiC papers, whose grit size ranged from 320 to 2,000.

4.2 Numerical set-up

In this section, the constitutive framework chosen in this work is introduced, followed by

a description of the FEM set-up and calibration.

4.2.1 Constitutive framework

The constitutive framework proposed in this work is based on a single-crystal plasticity

framework modified to account for twinning reorientation. It closely follows the work

of Staroselsky and Anand [158, 159]. For completeness, a summary of its formulation is

provided. The novelties of this new model are subsequently highlighted.

4.2.1.1 Crystal plasticity continuum formulation

The total deformation of a crystal is the result of three main mechanisms: dislocation

motion within the active slip systems, twinning motion within the active twin systems

and lattice distortion. Following Lee [209], the deformation gradient tensor F can be

uniquely decomposed following:

F = Fe · Fp (4.1)

where Fp and Fe are respectively the deformation gradient tensor accounting for the

cumulative effect of dislocation motion and shear due to twinning, and the elastic distortion

of the crystal lattice, with det(Fe) > 0 and det(Fp) = 1 (Figure 4.1). This distortion gives

rise to the Cauchy stress σ.

4.2. Numerical set-up 29

Figure 4.1: Multiplicative decomposition of the deformation gradient F.

The deformation power per unit reference volume is thus defined by:

P = P : F = Se : Ee + (Ce · Se) : Lp (4.2)

where

P = det(F)σ · F−T (4.3)

Se = det(Fe)Fe−1 · σ · Fe−T (4.4)

Ce = FeT · Fe (4.5)

Ee =1

2(Ce − I) (4.6)

Lp = Fp · Fp−1 (4.7)

are respectively the first Piola-Kirchhoff stress tensor, the second Piola-Kirchhoff stress

tensor relative to the relaxed configuration, the right Cauchy-Green elastic deformation

tensor, the Green-Lagrange elastic strain and the plastic velocity gradient tensor.

Assuming a linear elastic relation, as it is usually done in metallic single crystals, the

constitutive equation is given by [159]:

Se = C : Ee (4.8)

where C is a fourth-order anisotropic elasticity tensor.

By projecting Se on the slip system i and twin system α, the resolved shear stress in

systems i and α are then given by:

{

τ i = (Ce · Se) : Si0

τα = (Ce · Se) : Sα0

(4.9)

where{

Si0 = mi

0 ⊗ ni0

Sα0 = mα

0 ⊗ nα0

(4.10)

30

are the Schmid tensors for slip system i and twin system α in the reference configuration.

In this equation, slip (resp. twin) system i (resp. α) is characterized by a unit normal ni0

(resp. nα0 ) to the slip (resp. twin) plane, and a unit vector mi

0 (resp. mα0 ) denoting the

slip (resp. twin) direction in the reference configuration.

Extending Taylor [19] and Rice [210] approaches to include the twinning components,

the kinematics of dislocation of motion and twin shear can be described by:

Lp =∑

i

sign(τ i)γiSi0 +

∑

α

sign(τα)γαSα0 (4.11)

where γi (resp. γα) is the shear rate on slip (resp. twin) system i (resp. α). Subsequently

for simplification, superscripts i and j will refer to a slip system, and α and β to a twin

system.

Introducing si/α as the CRSS in system i/α, the conditions for slip and twinning, as

well as the associated flow rule, are written as:

φi = |τ i| − si ≤ 0

φα = τα − sα ≤ 0

γi/α ≥ 0, and γi/αφi/α = 0

(4.12)

Note that the unidirectionality of tensile twinning {1012}〈1011〉 is accounted for by

relaxing the absolute value condition on the resolved shear stress τα.

In Staroselsky and Anand [159], the flow rule was modified to include grain boundary

deformation mechanisms within a “grain boundary layer” surrounding each grain. This

feature was accounted for by adding an isotropic plasticity flow rule component to

Equation (4.7) weighted by a given grain boundary proportion. However, for the grain size

observed in the experimental samples (∼ 13 µm), it has been shown that grain boundary

deformation mechanisms are not significant at room temperature [69]. As a consequence,

this term was not included in the model presented here.

Finally, the evolution equations for slip and twin CRSS are given by a generalized

linear hardening relation between the CRSS and the shear rates of all systems:

si =∑

j

hij γj +∑

β

hiβ γβ

sα =∑

β

hαβ γβ +∑

j

hαj γj(4.13)

In this expression, hij , hiβ , hαβ and hαj are the hardening moduli accounting

respectively for the slip-slip, twin-slip, twin-twin and slip-twin interactions. In Staroselsky

and Anand [159], the moduli were taken equal to zero under the assumption that the

hardening mechanisms were exclusively due to grain boundary mechanisms. Here, the


moduli are chosen to follow power law type relations:

hij = qijhj, with

qij = (qsl + (1 − qsl)δij)

hj = h0;sl

(

1− sj

ssl

)asl

hαβ = qαβhβ , with

qαβ = (qtw + (1− qtw)δαβ)

hβ = h0;tw

(

1− sβ

stw

)atw

hiβ = h0;tw−sl

(

1− sβ

stw−sl

)atw−sl

hαj = h0;sl−tw

(

1− sj

ssl−tw

)asl−tw

(4.14)

where δij is the Kronecker symbol, where quadruplets (h0;sl, h0;tw, h0;tw−sl, h0;sl−tw),

(ssl, stw, stw−sl, ssl−tw) and (asl, atw, atw−sl, asl−tw) are respectively the reference self-

hardening parameters, the saturation stresses and the hardening exponents for (slip-slip,

twin-twin, slip-twin, twin-slip) interactions, and where (qsl, qtw) are the ratios of reference

cross-hardening parameter to reference self-hardening parameter for (slip, twin). Simple

power-law relations such as the ones proposed here in Equation (4.14) are commonly used

in crystal plasticity continuum models [132, 146, 147, 151, 155, 158, 163, 164]. Note that

other formulations using Voce-type law or more generally hyperbolic functions could also

be used; but all generally present the same drawback of being exclusively dependent on

the CRSS, occulting any dependence on the lattice defect population for example [132].

However, in the interest of limiting the number of parameters introduced in the current

model, the choice was made to use a simple power-law, while acknowledging the fact that

other models more physically grounded could potentially be used [211]. Future studies will

aim at studying the effect of the choice of the hardening model on the crystal deformation.

In the following, the CRSS will be defined by s0;sl and s0;tw for the slip sl and twin tw

system respectively, where sl ∈ {basal, pyr〈c+ a〉, pyr〈a〉, prism} for the basal, pyramidal

〈c+ a〉, pyramidal 〈a〉 and prismatic slips, respectively.

4.2.1.2 Crystal rotation

A total Lagrangian scheme is used for texture updating, see [158] for more details. Note

that such scheme can predict accurately the final texture but that intermediate textures

might not be predicted with the same precision.

By definition, shear by twinning is accompanied by a local rotation of the crystal. This

transformation is defined for HCP metals by the rotation matrix [158]:

Rα = 2nα ⊗ nα − I (4.15)

where nα is the current twin plane normal of the twin system α. This transformation

rotates the crystal such that the final orientation mirrors the inital one with respect to

the twin plane. Such change is of significant importance as a given crystal, once rotated

by nearly 90° as a consequence of twinning, might have an unfavorable orientation for

32

further twinning, thus switching the main crystal deformation mechanisms from twin to

slip driven mechanisms.

It is followed here the model proposed by Van Houtte [173]. The twin volume fraction

fα evolution in twin system α is characterized by:

{

fα =∫ t0 f

αdt

fα = γα

γ0

(4.16)

where γ0 is the twinning shear. A random number ζ ∈ [ζmin, ζmax], where ζmin and

ζmax represent the lower and higher bound for twinning event, is picked at a given time

t, and, within the chosen representative volume element (RVE) of an individual crystal,

fmax = maxα(fα) is compared to ζ. If fmax > ζ, the RVE is assumed fully rotated, the

crystal lattice (i.e. all current slip and twin systems normals and directions is rotated the

twin system α for which fα=fmax following Equation (4.15), and the volume fractions are

reinitialized to zero).

4.2.2 Finite element model

The constitutive model was used in the FE program ABAQUS/Explicit [212] by

implementing the previous constitutive law in a “user material” subroutine. In this sub-

section, the FE discretization of the polycrystal model is presented first, followed by the

model calibration.

4.2.2.1 Finite element discretization

Following the example of Staroselsky and Anand [159], the polycrystalline sample is

idealized as a stacked structure of 12×12×18=2,592 linear cubic elements with reduced

integration and hourglass control. Each element can represent a full crystal and is assigned

its own crystal lattice orientation. Such approach can present serious limitations when used

for studying microstructural evolution within a grain (Chapter 5), where multiple elements

are needed. However its flexibility constitutes a significant computational advantage for

simulation models aimed at studying the overall behavior of full polycrystalline samples,

such as in this work. The mesh is shown in Figure 4.2.

The experimental uniaxial compression tests (Section 4.1) were simulated by imposing

Dirichlet boundary conditions on the top and bottom surfaces of the model. Both

were constrained laterally to account for the absence of gliding during contact in the

experimental set-up, the top surface was displaced at a given velocity along the vertical

direction while the bottom one was fully constrained. The model being rate-independent

and the problem quasi-static, the actual compression velocity and the density could be

chosen arbitrarily so as to minimize inertial effects and computational calculation time.

The resulting simulation is thus fully quasi-static, which is a good approximation of the

very low rate of the experimental compression (10−4 s−1).


Figure 4.2: Uniaxial compression numerical tests along ND and RD. The resulting vonMises stress field is plotted for both cases.

Note finally that for all the simulations presented in this work, both spatial and

temporal convergences were verified.

4.2.2.2 Model calibration

Compression/double twinning The significance of compression and double twinning

in accommodating c-axis compression in Mg alloys is still not clear. Some studies have

recently reported the occurrence of compression and double twinning in polycrystalline

Mg alloys during c-axis compression [38,44,60,213,214]. However, traces of these types of

twins are only observed in very small regions, often very close to the fracture surfaces. It

has nevertheless been recently reported that, despite the low volume fraction of such twins,

their effect in the hardening of the Mg AZ31 alloy might be important [215]. On the other

hand, it is also widely accepted that pyramidal slip accommodates most of the compression

strain along the c-axis [30, 215]. In view of the controversial role of compression/double

twinning, we have chosen not to include this deformation mechanism in the continuum

model, as a first approximation.

Texture The model was calibrated against the stress-strain curves of Figure 4.4. In

the first case, the c-axis of the overall texture is on average aligned with the compression

direction, whereas, in the second case, it is perpendicular to it. Accordingly, the texture of

the mesh for the compression along RD is rotated by 90° in the mesh for the compression

along ND (Figure 4.2). The initial texture of the numerical model was finally tuned by

adding a gaussian noise (Figure 4.3) to the three Euler angles around each configuration

until an adequate match with the experimental poles figures was reached.

Note that both experimental pole figures seem to be more “diffuse” than the numerical

ones. However, the numerically modeled texture used in this work is chosen to be

34

(a) along ND (b) along RD

Figure 4.3: Comparison between the experimental (top) and the numerical (bottom) initialtexture for the two cases: (a) along ND; (b) along RD.

A B C D E F

0% 2% 4% 6% 11% 14%

Table 4.2: Strain at transition states.

representative of all samples in general. This explains the observed differences between

the pole figure for one given experimental sample and the one for the numerical model,

representative of all samples. Finally, the observed “splitting” in Figure 4.3, typical of

rolling textures in Mg alloys, has been previously attributed to the operation of pyramidal

slip during plane strain deformation [27] and was replicated in the models.

Material parameters The material parameters were chosen according to the following

scheme. Initial parameters were taken out of the literature [50, 51, 158, 159, 164, 216].

The ones that differed within these references were chosen so as to achieve the closest

fit with the experimental stress-strain curves. Finally, the other parameters (namely the

slip-twin/twin-slip hardening and twinning rotation parameters) were calibrated so as to

achieve the best fit while keeping the previous parameters fixed. The final set of parameters

is given in Table 4.3 and the resulting stress-strain curves corresponding to the uniaxial

compressions along ND and RD are shown in Figure 4.4.

For the purpose of analysis, the stress-strain curve for the compression along RD has

been divided into six regions. The deformation states at which one region transitions to

another are marked by A, B, C, D, E and F (Figure 4.4(b) and Table 4.2).


0 0.01 0.02 0.03 0.04 0.05 0.06 0.070

50

100

150

200

250

300

True strain

Tru

e s

tre

ss

(M

Pa

)

experiment

simulation

(a) along ND

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.160

100

200

300

400

True strain

Tru

e s

tress (

MP

a)

experiment

simulation

E FDCBA

(b) along RD

Figure 4.4: Experimental and simulated stress-strain curves (true) corresponding touniaxial compression of the Mg AZ31 alloy: (a) along ND; (b) along RD.

36

s0;basal s0;pyr〈c+a〉 s0;pyr〈a〉 s0;prism9 MPa 115 MPa 115 MPa 80 MPa

h0;sl h0;tw h0;sl−tw h0;tw−sl

600 MPa 80 MPa 0 MPa 1200 MPa

sbasal spyr〈c+a〉 spyr〈a〉 sprism15 MPa 190 MPa 190 MPa 150 MPa

s0;tw stw ssl−tw stw−sl

17.5 MPa 100 MPa n/a 100 MPa

asl atw asl−tw atw−sl

0.6 2 n/a 2

qsl qtw ζmin ζmax

4 1 0.75 0.8

γ0 C11 C12 C13

0.129 58 GPa 25 GPa 20.8 GPa

C33 C55

61.2 GPa 16.6 GPa

Table 4.3: Material parameters after calibration.

Note that the hardening related to the effect of slip on tensile twinning was suppressed

based on the relatively scarce (and sometime contradicting) information in the literature

on this interaction, and the observation that the region between the states of deformation

A and C (Figure 4.4(b))1 was found to be only weakly dependent on this parameter even

for large values of h0;sl−tw. Additionally, increasing this parameter would only steepen the

numerical curve away from the experimental one. The (ζmin,ζmax) parameters played an

important role in the C-to-D portion of the curve and the twin-slip hardening parameters

in the C-to-E portion. The final shape of both simulated samples after deformation with

the corresponding von Mises stress fields are shown in Figure 4.2.

Material parameters sensibility A unique set of slip systems will be activated [146].

However, it is not clear whether a calibration based on both considered stress-strain curves

should be enough to guarantee the uniqueness of the set of parameters. For example,

one would want to know whether a set of pyramidal hardening parameters along with

a set of twin hardening parameters could not be exchanged by two other sets without

noticeable differences in the stress-strain curves. To this end, an extensive parameter

sensibility study was done in which each individual parameter is varied by ±20%, ±50% or

±100% depending on the sensibility of the parameter. The strain at which the first major

noticeable influence of these variations on one of the portions of the RD/ND compression

stress-strain curves was then marked down and was reported in Table 4.4. Note that a

non-marked cell does not mean that there is no influence of the parameter but that a much

more important variability has been observed at a different strain (or on the other curve).

1This region should actually a priori be the one most directly influenced by these parameters (influenceof already existing basal slip on nascent twinning)

4.3. Results and discussion 37

Parameters RDA RDB RDC RDD RDE RDF ND

Twin

s0;tw × ×stw ×h0;tw ×atw × ×qtw × ×

Twin Rotationζmin × ×ζmax × ×

Twin-Slipstw−sl × ×h0;tw−sl × ×atw−sl × ×

Slip

s0;pyr × × ×spyr × ×

s0;prism ×sprism ×s0;basal ×sbasal × ×h0;sl ×asl ×qsl ×

Table 4.4: Material parameters sensibility.

Table 4.4 shows that the twinning, twin-slip interaction and pyramidal slip parameters

are successive quasi-independent mechanisms in the compression along RD. Basal and

prismatic slip influences are on the other hand overlapping independently some of the

zones of influence of the other mechanisms, but basal slip parameters do also influence

majoritatively the ND compression. Overall, a proper fit of the stress-strain curves at

states A, B, C, D, E and F as well as for the ND compression curve can be done

with a quasi-independent calibration of twinning, pyramidal and prismatic parameters

majoratively for the RD compression case, and pyramidal and basal for the ND case (with

cross-calibration for pyramidal), in this order. Additionally, the three main mechanisms

(twinning, basal and pyramidal slips) are shown to influence in a complementary manner

the stress-strain curves.

4.3 Results and discussion

In the previous section, the model was calibrated against two uniaxial compression

tests with one unique set of material parameters, validating the ability of the model to

describe the Mg AZ31 alloy deformation mechanisms under both slip- and twin-dominated

deformations. These deformation mechanisms are identified in the following by studying

texture evolution, slip/twin activities and the related controversy on pyramidal 〈c+ a〉

splitting. Finally, a failure criterion based on slip activity is proposed.

38

Figure 4.5: Comparison between {0001} experimental and numerical textures at deforma-tion state A, B, C, D, E and F for the compression along RD.

4.3.1 Texture evolution and slip/twin activities

The {0001} pole figures at the experimental and simulated deformations states A, B, C, D,

E and F for the compression along RD are illustrated in Figure 4.5. Figure 4.6 shows the

normalized overall slip/twin activity vs. the overall strain for both uniaxial compression

cases.

The first observation is that in both cases, basal slip is the overall most important

mechanism. After the initial yielding, the A-to-B region is mostly driven by twinning

despite some basal slip activity. This basal activity is due to the systems that are slightly

tilted and thanks to the very low basal CRSS. The initial presence of twinning is confirmed

by the clear appearance of twin driven lattice rotation, as can be seen in the experimental

{0001} pole figure at state B (Figure 4.5), followed by the sudden drop of its activity

(Figure 4.6(b)).

At state C, the experimental pole figure shows that all the crystals have rotated. It

appears that the simulated results are delayed in this rotation, rotating slowly until slightly

after state D. This delay in the pole figures can be explained by two factors:

� Firstly, it has been experimentally observed that the boundaries of tensile twins

migrate very rapidly [215]. However, the model, because of its one-to-one association

between grain and element, precludes a “gradual rotation” by twin growth within

the grains, making it a sudden switch from one crystalline orientation to another

one (Sub-section 4.2.1.2). This is actually confirmed by comparing the twinning

rotation evolution for the simulation and the experiments (Figure 4.7). In this figure,

the experimental volume fractions of twinned regions in the samples compressed


0 0.01 0.02 0.03 0.04 0.05 0.06 0.070

0.2

0.4

0.6

0.8

1

True strain

No

rma

lize

d s

lip

/tw

in a

cti

vit

y

Basal

Pyramidal <c+a>

Pyramidal <a>

Prismatic

Twin

(a) along ND

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.160

0.2

0.4

0.6

0.8

1

True strain

No

rmali

zed

sli

p/t

win

acti

vit

y

Basal

Pyramidal <c+a>

Pyramidal <a>

Prismatic

Twin

E FCBA D

(b) along RD

Figure 4.6: Normalized slip/twin activity for both cases: (a) compression along ND; (b)compression along RD.

40

0 0.02 0.04 0.06 0.08 0.1 0.12 0.140

20

40

60

80

True Strain

Ro

tate

d G

rain

s %

experiment

experiment

simulation

simulation

~ 4.5%

Figure 4.7: Experimental and numerical twinning rotation fraction evolution.

uniaxially along RD were calculated at various strain levels from the X-ray ODF

data using the software MTEX. A misorientation of ±30° was used (this value was

identified by means of the orientation spread in the ODF)2. As can be seen, the

inflections of the curves are very similar, but there is a delay of roughly 4.5%.

One more time, this delay is explained by the binary-type rotation of the model,

as opposed to gradual. In other words, the simulation curve on Figure 4.7 is

representative of the averaged fraction of fully rotated grains, and not of the averaged

fraction of rotated lattice within each grain. This restriction of the model will be

solved in future works by considering multiple elements per grain and will be further

discussed in the next chapter.

� Secondly, whereas the experimental pole figures are taken by considering a thin slice

of materials within the bulk, the simulated ones include the whole polycrystal, and,

as can be seen on Figure 4.8, the non-sliding boundary conditions are responsible

for a twin rotation gradient close to the top and bottom surfaces. This restriction

thus leads to an observed delay in the twin rotation when averaged over the whole

sample. For example, at 4% deformation, 13% of all the grains are rotated, but

the outer and center element layers respectively have 1% and up to 27% of rotated

grains (Figure 4.9).

During the rotation from C to D, basal slip activity is slightly increasing before

regaining its previous level after full rotation. Prismatic and pyramidal 〈a〉 activities

are also increasing during this change of crystal orientation, but overall remain relatively

small.

Finally, the D-to-E region (followed by the E-to-F region, where all the main defor-

mation mechanisms saturate) exhibits a transition to pyramidal 〈c+ a〉 slip dominated

2Note that the twinned volume fraction was alternatively calculated in the sample deformed up to4% by EBSD, showing similar results, however the technique described above being more flexible andtime-efficient for post-processing, it was ultimately chosen for this study


(a) (b) (c) (d) (e) (f)

Figure 4.8: Twinning rotation state in a longitudinal cut of the sample (blue: non-rotated,red: rotated) for the states of deformation A, B, C, D, E and F.

0 2 4 6 8 10 12 14 16 180

5

10

15

20

25

30

Element layer

Ro

tate

d G

rain

s %

Figure 4.9: Twinning rotation fraction as a function of the vertical element layer in thesample at 4% deformation.

42

deformation (Figure 4.6(b)). Such change is explained by the polarity of twinning, pro-

hibiting its activity during compression when the c-axis is aligned with the compression

direction. As a consequence, the next most favorable deformation mechanism is pyramidal

〈c+ a〉 slip (with basal slip present during the whole deformation). This is confirmed in

Figure 4.6(a) where the crystals have the c-axis aligned with the compression axis from the

beginning. In this case, basal slip is dominant at the beginning of the deformation, and

when blocked by the misorientation of the neighboring grains, is transitioned to pyramidal

〈c+ a〉 slip driven deformation. Note finally that all systems did not rotate at stages E

and F as it did experimentally. This delay can be explained by the same factors that

delayed the onset of rotation.

The analysis above shows that, despite its high room temperature CRSS, pyramidal

〈c+ a〉 slip operates in conjunction with basal slip during room temperature compression

of the rolled Mg AZ31 alloy sheet along RD and ND (Figure 4.5). The activation of

these two slip systems explains the texture development as well as the corresponding

stress-strain curves (Figure 4.4). This is consistent with the findings of Yi et al. [59] who

determined the relative slip/twin activities during tension of a Mg AZ31 alloy extruded

bar perpendicularly to the extrusion axis by VPSC modeling, and found that pyramidal

〈c+ a〉 slip does play an important role under such conditions. In their case, however,

the contribution of pyramidal 〈c+ a〉 slip is smaller with respect to the one of basal slip

due to the wider spread of basal planes around the compression axis in their extruded bar

than in the rolled sheet investigated here. Wang et al. [217] also recently calculated the

slip/twin activity during uniaxial deformation of a rolled Mg AZ31 alloy sheet using several

VPSC models. More specifically, they fitted different models to experimental tension and

compression tests along RD and used the same parameters to predict compression along

ND. They concluded that the affine VPSCmodel gives the best overall performance. When

compressing along ND, in both the affine VPSC model and the one presented here, only

basal and pyramidal slips operate. However, the cumulative activity of pyramidal slip

predicted by the affine VPSC model is significantly smaller than the one obtained in the

present work. When compressing along RD, both models predict the activation of tensile

twinning and basal slip in the first stages of deformation, but the relative contribution

of twinning is much smaller when using the affine model than in the observed prediction.

In the last stages of deformation both models predict the activation of both basal and

pyramidal slip. Again, the affine VPSC model predicts a larger contribution of basal

slip. In summary, similar deformation modes are predicted to operate using both the

affine VPSC model and the current CPFE model. However, in all cases, the VPSC

model estimates a higher relative contribution of basal slip. This might be due to the

more diffuse basal texture but such differences certainly highlight the necessity for further

comparisons between different numerical models, where a priori acceptable simplifications

might possibly lead to erroneous description of the deformation mechanisms. Finally, the

activity of pyramidal 〈c+ a〉 slip during the last stages of deformation of a rolled Mg AZ31

alloy sheet compressed along RD at 200 � has also been predicted by Choi et al. [164]


using a similar model to the one presented in this work. This finding is, however, not very

surprising, as it is well known that the CRSS for pyramidal slip decreases rapidly with

temperature and therefore its activity increases significantly at temperatures higher than

approximately 180 �.

4.3.2 Fracture mechanisms

An analysis of the sample fracture surface suggested that the material failure is stemming

from intragrain semiductile failure (Appendix A). This could a priori indicate the

formation of a shear zone eventually leading to an intragrain failure. In order to confirm

or infirm this hypothesis, a criterion based on the cumulated average slip activity for slip

system i at time t was defined:

Γi =

∫ t

t0

< γi > dt (4.17)

where < γi > is the average slip rate at time t over the full sample, and t0 the time

reference. Note that such criterion is purposefully macroscopically averaged at the sample

level. Defining a criterion at the grain level (thus defining a failure initiation criterion,

see for example Jerusalem et al. [150]) in a model such as here, where grains are modeled

by only one element, would actually miss the purpose of the proposed criterion (defining

overall sample failure).

It has been shown that a dislocation interacting with a twin boundary during twin

growth is not transferred through the twin boundary, but instead is decomposed into

interfacial defects [218]. An intragrain failure criterion based on slip accumulation thus

requires that only the slip activity since the time of the last twin lattice rotation (if any)

should be taken into account. Note that such consideration does not concern the hardening

model, based on the assumption that previous dislocations, even possibly dismantled into

other stacking faults, still participate to the lattice hardening. However the bulk of the

crystal itself does not a priori contain full residual dislocations after the passing of the

twin boundary [218], and its relation to intragrain failure should be reinitialized. The

time at which the stress-strain curve (Figure 4.4(b)) regains a quasi-elastic behavior after

the rotation of the crystals (ǫ(t0) = 6%) was chosen as the “reinitialization time” t0 (in

the ND case, as very little twinning occurs, t0 = 0). Figure 4.10 shows the evolution

of Γi vs. time for all slip systems for both cases (including the reinitialization after twin

rotation when needed) up till experimental fracture. As could be expected from Figure 4.6,

prismatic and pyramidal 〈a〉 slip activity seem highly uncorrelated with a potential slip

related failure mechanism. On the other hand, the cumulated average basal and pyramidal

〈c+ a〉 activity reach the same value of ∼ 0.04 and ∼ 0.015 respectively.

These results alone cannot permit to conclude that basal cleavage or pyramidal 〈c+ a〉

cleavage, is responsible for the material failure. However, Mg alloys fracture mode under

the conditions of temperature (0.33 Tm) and normalized stress (σ/E ∼ 6×10−3) examined

in the present paper is of the mixed mode type “cleavage 3/ductile” or “semibrittle

44

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.160

0.01

0.02

0.03

0.04

0.05

True strain

Co

rre

cte

d c

um

ula

ted

av

era

ge

sli

p/t

win

ac

tiv

ity

Basal RD

Basal ND

Pyramidal<c+a> RD

Pyramidal <c+a> ND

Pyramidal <a> RD

Pyramidal <a> ND

Prismatic RD

Prismatic ND

Twin RD

Twin ND

Figure 4.10: Corrected cumulated average slip/twin activity for both cases of compression:along RD and along ND.

intergranular fracture” [219], as substantial plastic strain precedes cleavage. The cleavage

planes vary for different HCP metals. Yoo calculated the ratio between the critical fracture

stresses in basal and prismatic planes for a number of HCP metals [17]. He found out

that basal planes are clearly the most preferred cleavage planes in Zn [17, 220], whereas

prismatic planes are favored in titanium and zirconium. Basal and prismatic planes are,

respectively, those with the smallest CRSS in those metals. Mg is presented by Yoo [17]

as an intermediate case. However, since basal slip is the most active deformation mode,

it seems logical that cleavage is favored in basal planes. In fact, it has been reported

earlier [20] that rolled Mg sheet at room temperature fail along bands formed by grains

in which the basal planes are parallel to the band plane. The easiness of cleavage along

basal planes has also been verified in an Mg AZ31 alloy by Somekawa et al. [58]. It thus

can conclude that basal cleavage is reponsible for the material failure.

As a conclusion, these results suggest: a) that sample failure is directly related

to an excess of basal activity in agreement with experimental observations, probably

primarily due to the formation of a shear zone along the 〈0001〉 direction leading eventually

to material failure: b) that when the compression axis is preferably oriented along

ND, pyramidal 〈c+ a〉 and basal slip activities follow a very similar pattern, working

collaboratively in the same proportion; c) and finally, that the final failure is in any case

independent of the other slip system activity (pyramidal 〈a〉 and prismatic) and more

importantly of the twin activity.

4.3.3 Validation

In the following, the previous study is complemented by using the model on a third

configuration: plane strain compression along RD, constraining along TD. To this end,

a specific die made of hardened stainless steel was designed for such purpose. In order


to minimize friction effects, the sample surfaces were lubricated using Teflon tape. The

deformation rate was identical to the previous tests (10−4 s−1).

The experiments showed a relatively important variability and a total of four (instead

of three) tests was used for the final averaged stress-strain curve. The experimental stress-

strain curve (with its errorbars) and the numerical one are given in Figure 4.11. The four

stages used in this figure are given in Table 4.5.

0 0.02 0.04 0.06 0.08 0.1 0.120

100

200

300

400

True Strain

Tru

e S

tre

ss

(M

Pa

)

Errorbar Experiment

Experiment

Simulation

A B C D E

Figure 4.11: Experimental and numerical stress-strain curves for the plane straincompression along RD, constrained along TD.

A B C D E

0% 2.5% 4.5% 6.5% 10.3%

Table 4.5: Transition strain between regions for the plane strain compression case.

The two stress-strain curves both present an earlier point of inflection, higher hardening

rates and higher stress values when compared to their respective uniaxial RD compression

counterpart (Figure 4.4). Additionally, the two curves seem to deviate between stages C

and E, incidentally corresponding to the portion of the experimental curve with the most

important variability. In view of this, the simulation captures the experimental behavior

relatively well.

4.3.3.1 Texture evolution and slip/twin activities

The comparison of the {0001} texture between experiment and simulation for these

different stages, and the overall slip/twin activity are given in Figures 4.12 and 4.13,

respectively.

The pole figure evolution is relatively well caught by the simulation, with the same

delay in the crystal rotation as was previously observed (Figure 4.5 and Sub-section 4.3.1).

Furthermore, even accounting for the different strain-to-failure, a comparison between

Figures 4.6(b) and 4.13 exhibits a very similar evolution of twin and basal slip activity

(as well as pyramidal 〈a〉 which remains almost inexistent), but weaker contribution of

46

Figure 4.12: Comparison between {0001} experimental and numerical textures atdeformation state A, B, C, D and E for the plane strain compression along RD, constrainedalong TD.

0 0.02 0.04 0.06 0.08 0.1 0.120

0.2

0.4

0.6

0.8

1

True strain

No

rmalized

sli

p/t

win

acti

vit

y

Basal

Pyramidal <c+a>

Pyramidal <a>

Prismatic

Twin

A B C D E

Figure 4.13: Normalized slip/twin activity for the plane strain compression along RD,constrained along TD.

4.4. Conclusion 47

0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12 0.130

0.01

0.02

0.03

0.04

0.05

True strain

Co

rre

cte

d c

um

ula

ted

a

ve

rag

e s

lip

/tw

in a

cti

vit

y

Basal

Pyramidal<c+a>

Pyramidal <a>

Prismatic

Twin

Figure 4.14: Corrected cumulated average slip/twin activity for the plane straincompression along RD, constrained along TD; the dash line corresponds to the acceptablerange of values for the strain-to-failure.

pyramidal 〈c+ a〉 and earlier contribution of prismatic, implying that prismatic slip is

taking over part of the pyramidal 〈c+ a〉 contribution under plane strain conditions, when

compared to uniaxial.

4.3.3.2 Fracture mechanisms

Finally, the fracture criterion proposed in Sub-section 4.3.2 is tested here for the

new configuration. The corrected cumulated average slip/twin activity based on the

methodology described earlier is shown in Figure 4.14.

Note that the experimental curves were averaged up to the smallest strain-to-failure

(10.4%). However, because of the important variability of the experimentally observed

strains-to-failure, which here were spanning a total of 2.6% (10.4%, 10.9%, 11.2%, 11.6%

and 13%), the simulation could possibly be extended up to 13%. As a consequence, the

results in Figure 4.14 were run up to 13%.

As hinted by Figure 4.13, Figure 4.14 exhibits a lower pyramidal 〈c+ a〉 cumulated

average activity than in Figure 4.10, but a higher prismatic activity. The basal activity,

on the contrary, is very similar. More importantly, the failure cumulated average basal

activity threshold of ∼ 0.04, as defined in Sub-section 4.3.2, is reached within the strain-

to-failure range of 10.4%-13%. On the contrary, the pyramidal 〈c+ a〉 threshold of 0.015 is

not reached. Based on the previous discussion, this further validates the fracture threshold

based on basal activity defined and discussed earlier.

4.4 Conclusion

A CPFE model aimed at fully describing the intrinsic deformation mechanisms between

slip and twin systems has been developed. An experimental campaign consisting of a set

48

of uniaxial and plane strain compression tests at room temperature and quasi-static strain

rate, as well as an exhaustive analysis and examination of the crystallographic texture

at different stages of deformation of the Mg AZ31 alloy have been carried out. The

model was then calibrated and validated against these tests. The flexibility of the overall

model was demonstrated by casting light on an experimental controversy on the role of

the pyramidal 〈c+ a〉 slip vs. compression twinning in the late stage of polycrystalline

deformation. Finally, a macroscopic failure criterion based on basal activity was shown to

accurately describe the failure of materials in what would thus be a shear zone along the

〈0001〉 direction eventually leading to material failure. This criterion seems furthermore to

be independent of the twin activity but tightly linked to pyramidal 〈c+ a〉 and prismatic

activities.

CHAPTER 5

3D investigation of grain boundary-twin interactions in a Mg AZ31 alloy

by electron backscatter diffraction and continuum modeling

This chapter presents a combined 3D experimental and modeling approach to gain a better

understanding on twin propagation and growth mechanisms in a Mg AZ31 alloy. The 3D

microstructure characterization was performed by the 3D EBSD technique. The role of

GB misorientation on twin transfer is investigated by analysis of the local plastic 3D state

at the GB interface. Twin growth is evaluated by characterization the 3D twin morphology

and the corresponding crystallographic dependence of the Schmid law. The CPFE model

developed in the previous chapter is extended to include the topological information of

individual grains.


The material studied is a 3 mm-thick sheet of a Mg AZ31 alloy, rolled and annealed, with

an average grain size of ∼ 25 µm and a typical basal texture. As seen earlier, during

compression along RD, twin nucleation and growth proceed readily to the extent that, at

strains of 5%, the tensile twin volume fraction is larger than 50% [88], and many twins have

consumed the entire volume of the original grains. With the aim of investigating the role

of GB on twin transfer, 3 mm×3 mm×4.5 mm prisms (long axis along RD) were machined

and tested in compression along RD at room temperature and at strain rate of 10−4 s−1

in an electromechanical Servosis ME 405/10 machine up to strains of 1.6% and 2.6%.

These small strain levels were chosen with the aim of characterizing the microstructure at

a deformation stage where twin growth is still limited, thus allowing for the evaluation of

twin nucleation and growth events.

The microstructure of the tested samples was investigated along three perpendicular

planes, using 2D EBSD on a FEG-SEM JEOL 6500F equipped with an EDAX/TSL

49

50

system. These measurements allowed for the estimation of twin volume fraction variations

with strain along the three planes of observation, i.e. RD, ND and TD, and the

evaluation of the influence of GB misorientation on the frequency of ATPs. Sample

preparation for EBSD included grinding with SiC papers of grit size ranging from 1,200

to 4,000, mechanical polishing with diamond slurries of 6 µm, 3 µm and 1 µm, and

final electrochemical polishing for 45 s at 20 V, using the AC2 commercial electrolyte.

The microstructure of the sample strained to 2.6% was further analyzed by 3D EBSD-

based orientation microscopy, using serial sectioning with a (FIB)-FEG-SEM dual beam

instrument (Zeiss Crossbeam 1530) [192]. One volume consisting of 50 sections of

dimensions 23 µm×25 µm×100 nm was milled using a 30 kV Ga+ ion beam. Qube was used

for volume reconstruction [194]. Significant efforts were made to select an ideal volume

containing several boundaries with different misorientation angles. The determination of

the active twin variants was carried out by the method of the minimum deviation angle

(θdev) [92]. This method consists of the following (Appendix B): first, the orientation

matrix of each original grain gPiis calculated from the corresponding Euler angles; next,

the six possible twin rotations are applied to each gPito obtain the orientation matrices

corresponding to the six variants gTij; finally, the θdev values between the six calculated

gTijand the orientation matrix of the twins observed experimentally are estimated, and

the smallest θdev is identified as corresponding to the active twin variant.

5.2 Numerical set-up and validation

This work is an extension of the CPFE model previously developed for the Mg AZ31 alloy

in Chapter 4. This extension aims at capturing more realistic polycrystal features by

considering the topological information for the grains. In this model, the polycrystalline

aggregate is represented as a 3D Voronoı tessellation with a large number of elements per

grain, thus allowing for the high resolution study of the local intragranular mechanical

fields. The extended 3D polycrystalline model, whose volume is 1 mm3, consists of 2,592

grains, each one with its own crystallographic orientation, with an overall texture matching

the experimental one. The new mesh, generated using the Neper open-source software

package [221], consists of a total of 475,947 linear tetrahedral elements (Figure 5.1). The

constitutive framework as well as the set of material parameters were taken from the

previous chapter (Table 4.3).

The robustness of the new model presented here was first validated against experiments

and against the previous predictions of the original model described in Chapter 4. The ex-

perimental campaign carried out for model validation consisted of several compression tests

along RD and ND, performed at 10−4 s−1 and room temperature up to different strains

ranging from 2% to 14%. Compression specimens were 3 mm×3 mm×4.5 mm prisms and

3 mm×3 mm×3 mm cubes for the RD and ND tests, respectively. Figures 5.2(a) and (b)

reveal the good agreement reached between two simulated and experimentally measured

true stress-strain curves and the normalized slip/twin activities of both numerical models

5.2. Numerical set-up and validation 51

Figure 5.1: Polycrystalline aggregate mesh of the proposed extended model.

(at the specimen center). Figure 5.3(a) illustrates the evolution of the cumulative twin

volume fraction with strain, measured experimentally by integration of the appropriate

texture components of the X-ray ODF and simulated using both models. Experiments re-

veal an increase in the twin fraction at the early stages of deformation (ǫ < 4%). Although

both models are unable to fully capture the progressive rotations associated with twinning,

the extended model proposed here agrees better with the experimental data. Moreover,

the predictions of the extended model seem to be in good agreement with experiments

reported recently in the same alloy with a different microstructure [222]. Figure 5.3(b)

illustrates the simulated twin volume fraction measured along three planes perpendicular

to RD, ND and TD in the center of the sample. Only a slight difference is observed be-

tween the three curves, in agreement with Ref. [223], which revealed that relevant twin

statistics obtained from 2D EBSD maps at a given specimen location are independent of

the viewing direction. These small discrepancies can be attributed to the geometry of

the twins, depicted in Figures 5.4(a) and (b). These figures illustrate two experimental

volumes reconstructed using several 2D EBSD maps obtained along planes perpendicular

to the RD, ND and TD in the Mg AZ31 alloy sample compressed along RD up to strains

of 1.6% and 2.6%, respectively. The maps were obtained ∼ 1 mm deep below the surface

in each case, implying that they correspond to different locations in the specimen. It can

be seen that a smaller twin fraction (< 10%) is observed in planes perpendicular to the

test direction (RD). These findings are fully captured by the simulations (Figure 5.4(c)),

where the model boundary conditions seem to lower drastically the activity of twinning in

the same locations (Movie 01 in the Supplementary Material on CD-ROM). The present

results are, furthermore, consistent with previous experiments and modeling of twinning

in pure zirconium [223].

Overall, the mesoscopic behavior of the extended model proposed here matches the

averaged mesoscopic behavior of the initial model (Figure 5.1). As the latter was carefully

calibrated against the experimental results in Chapter 4, the new model can thus be

52

0 0.02 0.04 0.06 0.08 0.1 0.12 0.140

50

100

150

200

250

300

350

True strain

Tru

e s

tre

ss

(M

Pa

)

(ND) experiment

(ND) initial model

(ND) extended model (Voronoï)

(RD) experiment

(RD) initial model

(RD) extended model (Voronoï)

(a)

0 0.02 0.04 0.06 0.08 0.1 0.12 0.140

0.2

0.4

0.6

0.8

True strain

No

rmali

zed

sli

p/t

win

acti

vit

y

Basal

Pyramidal <c+a>

Pyramidal <a>

Prismatic

Twin

(b)

Figure 5.2: (a) Comparison of experimental and simulated true stress-true strain curvescorresponding to uniaxial compression along RD and ND; (b) normalized slip/twinactivities for both numerical models (dashed line corresponds to initial model describedin Chapter 4).

5.2. Numerical set-up and validation 53

0 0.02 0.04 0.06 0.08 0.1 0.12 0.140

20

40

60

80

100

True Strain

Tw

inn

ed

vo

lum

e f

rac

tio

n (

%)

experiment

initial model

extended model (Voronoï)

(a)

0 0.02 0.04 0.06 0.08 0.1 0.12 0.140

20

40

60

80

100

True Strain

Tw

inn

ed

are

a f

racti

on

(%

)

Plane perpendicular to ND

Plane perpendicular to RD

Plane perpendicular to TD

(b)

Figure 5.3: (a) Comparison of the cumulative twin volume fraction measured experimen-tally and predicted by the initial model described in Chapter 4 and by the extended model;(b) twin volume fraction predicted by the extended model along three planes perpendicularto RD, ND and TD in the center of the polycrystalline aggregate.

54

Figure 5.4: Qualitative estimation of the twin volume fraction along three orthogonalplanes in the Mg AZ31 alloy tested in uniaxial compression along RD: (a) volumereconstructed from three 2D EBSD maps measured in the sample deformed up to 1.6%;(b) volume reconstructed from three 2D EBSD maps measured in the sample deformed upto 2.6%; (c) twinned regions (dark) predicted by the extended model in a sample deformedup to 5%.

considered equally calibrated. The similarities and differences between both models are

discussed later. Finally, both spatial and temporal convergences were checked for all

simulations.

5.3 Results and discussion

5.3.1 3D variant analysis

Figure 5.5 illustrates the 3D reconstruction of the inverse pole figure maps along RD

obtained by the 3D EBSD technique. The volume dimensions are 5 µm in TD (height,

blue axis), 25 µm in RD (length, green axis) and 23 µm in ND (width, red axis). P1,

P2, P3, P4 refer to the four original grains, and Tij to variant j of grain i. Tables 5.1(a-

d) list the SF corresponding to the six tensile twin variants for each of the four grains

analyzed. The Euler angles (ϕ1,Φ, ϕ2) of all grains and twins analyzed are summarized in

Table 5.1. The minimum θdev approach [92] was used to identify the active twin variants.

This method provides conclusive evidence on the nature of the variant pair activated, i.e.

it allows differentiation between primary, secondary and tertiary variants. The θdev values

corresponding to each of the Tij twins are listed in Tables 5.1(a-d). The minimum θdev

values associated with the active variants are highlighted in bold letters. It can be noted in

Table 5.1 that, within a specific variant pair, the active variants do not always correspond

to those with the highest SF. This can be due to either some ambiguity inherent to the

deviation angle method, which does not take into account the small rotations that might


Figure 5.5: Sample compressed along RD up to 2.6%. 3D reconstruction of inverse polefigure maps on the RD obtained by the 3D EBSD method. P1, P2, P3, P4 refer to the fouroriginal grains, and Tij to variant j of grain i.

occur due to the simultaneous activation of crystallographic slip, or the fact that the

effective stress state at the grain core may deviate slightly from the macroscopic applied

stress.

In the following, the twin variants present in the four grains of the reconstructed 3D

volume are analyzed. Grain P1 is favorably oriented for twinning and contains three twin

variants: T11 is a primary variant (SF = 0.4759), T12 a secondary variant (SF = 0.1257),

and T13 a tertiary variant (SF = 0.0981). Grain P2 is also favorably oriented for twinning

and, at least within the volume analyzed, contains only one primary variant with a very

high SF (T21, SF = 0.4854). Grain P3 has an orientation that is less favorable for tensile

twinning. In this grain, two variants are active, both with moderate to very low SF: one

primary variant (T31, SF = 0.2074) and one secondary variant (T32 = 0.0486). Grain

P4, unfavorably oriented for tensile twinning, contains one primary variant (T41) with a

moderately low SF (SF = 0.1759).

The morphology of tensile twins, examined using mostly 2D microscopy techniques

such as OM and 2D EBSD, has been commonly described as plate, disk or needle-like [78].

Consistently, it has been reported that the shape of a twin boundary is generally expected

to be either a doubly convex or a planar-convex lens [26]. Thus, models usually consider

twins as lamella of constant thickness or ellipsoidal inclusions [48, 174, 224]. The present

study suggests that twin variant morphology may be related to the corresponding SF.

Figure 5.6 illustrates the 3D geometry of several twin variants. In particular, variant T11

is depicted in Figure 5.6(a). As inferred from 2D observations, the morphology of this

twin, which has a very high SF, resembles that of a plate, with approximately constant

thickness. It spans the entire volume of the scanned grain. Figure 5.7(a) illustrates the

56

(a)

P1: (87.6, 85.8, 277);

T11: (179.4, 83.3, 205.8); T12: (0.2, 36.8, 329.9); T13: (193.9, 24.3, 193.2)

Twin variants SF θdev(T11) θdev(T12) θdev(T13)

(1012) [1011] -0.4759 2.41 60.14 59.72(1012) [1011] -0.4914 4.99 60.65 60.02(1102) [1101] -0.1257 59.85 1.35 60.61(1102) [1101] -0.1339 60.01 8.62 60.29(0112) [0111] -0.0981 60.24 60.47 0.98(0112) [0111] -0.1055 60.37 59.89 6.47

(b)

P2: (81.3, 76.2, 266.4); T21: (354.7, 85.8, 343.6)

Twin variants SF θdev(T21)

(1012) [1011] -0.4854 0.69(1012) [1011] -0.4777 7.99(1102) [1101] -0.2023 60.33(1102) [1101] -0.1973 60.80(0112) [0111] -0.0601 59.68(0112) [0111] -0.0574 60.16

(c)

P3: (268.5, 89.2, 120.9);

T31: (4.2, 58.1, 24.5); T32: (181.7, 61.7, 149.5)

Twin variants SF θdev(T31) θdev(T32)

(1012) [1011] -0.2641 8.38 59.57(1012) [1011] -0.2074 2.34 59.18(1102) [1101] 0.0194 61.70 6.94(1102) [1101] 0.0486 61.34 1(0112) [0111] 0.0316 59.34 60.82(0112) [0111] 0.0591 58.72 61.15

(d)

P4: (316.9, 50.8, 41); T41: (193.1, 50.6, 137.6)

Twin variants SF θdev(T41)

(1012) [1011] -0.1160 7.96(1012) [1011] -0.1759 2.46(1102) [1101] 0.1454 59.54(1102) [1101] 0.1255 58.95(0112) [0111] 0.0500 61.40(0112) [0111] 0.0101 61.12

Table 5.1: SF and θdev corresponding to all the twins contained in the volume of Figure 5.5;the values of the Euler angles (ϕ1,Φ, ϕ2 – in degrees) corresponding to the original grains(Pi) and the twins (Tij) are included. The minimum θdev values are highlighted in boldletters.


(a)

(b)

(c)

Figure 5.6: 3D morphology of different twin variants: (a) T11 (volume size: 5 µm in TD(blue axis)×10 µm in RD (green axis)×15.8 µm in ND (red axis)); (b) T12 (orange) andT13 (yellow) (volume size: 5 µm in TD×16.2 µm in RD×11 µm in ND); (c) T41 (volumesize: 5 µm in TD×8.8 µm in RD×8.2 µm in ND).

58

0 1 2 3 4 51

1.5

2

2.5

3

3.5

4

Depth (µm)

Ma

xim

um

tw

in t

hic

kn

es

s (

µm

)

T111

T112

T113

T114

(a)

0 1 2 3 4 50

1

2

3

4

5

6

Depth (µm)

Maxim

um

tw

in t

hic

kn

ess

(µ

m)

T12

T13

(b)

Figure 5.7: Maximum twin thickness measured at different heights in grain P1: (a) foreach T11 lamella (the four lamella are named T111, T112, T113 and T114); (b) for secondary(T12) and tertiary (T13) twins.

variation in the maximum thickness of T11 lamella throughout the grain interior. This

plot confirms that the maximum twin thickness is almost constant throughout the grain.

Variants T21, which also have very high SF, are observed to have a morphology similar

to that of T11. Figures 5.6(b) and (c) illustrate the 3D geometry of variants T12, T13

and T41. Two rotated images of each twin variant are presented in order to facilitate the

observation of the 3D structure. These variants, which have small SF (ranging from 0.0981

to 0.1759), are irregularly shaped. In order to assess the validity of 2D estimations of the

twin thickness, the maximum thickness of the T12 and T13 variants at each 2D section

of the analyzed 3D volume is represented in Figure 5.7(b). Large variations are clearly

visible. This figure indicates that twin thickness estimations from 2D characterization

tools may lead to significant deviations from the 3D twin thickness.

The above observations may be rationalized as follows. It has been widely reported

that, during tension or compression loading of single crystals and polycrystals, the first


twins to form are primary variants, which have the highest SF [26, 174]. At such early

stages of deformation, the stress state at the core of the grains that are favorably oriented

for twinning can be assumed to be relatively homogeneous. Accordingly, one can expect

that primary variants, which have a large tendency to grow [89], adopt a regular (plate-

like) shape, related to the orientation of the uniaxial macroscopic stress. The strain

accommodation by primary variants is, however, limited, and local perturbations due

to the presence of the original GB and of the newly created twin boundaries lead to

intragranular effective stresses that may differ significantly from the macroscopic applied

stress state [224]. Non-Schmid effects, such as the activation of secondary and tertiary twin

variants, have been attributed to the presence of such stress fluctuations [48, 75, 76, 92].

It is therefore suggested that the irregular morphology of secondary and tertiary variants

observed here is related to the heterogeneous nature of the local effective stress field.

Furthermore, it is known that stress fields develop in the vicinity of twin boundaries [224]

and, thus, the presence of the irregularly shaped boundaries of non-primary variants can

be expected to further increase the heterogeneity of the intragranular local mechanical

fields. These findings indicate that twin variant morphology characteristics should be

addressed in models aimed at providing a full description of the mechanical behavior of

Mg alloys and other HCP metals.

5.3.2 Effect of GB misorientation on twin transfer

It has been reported recently that twin nucleation in Mg alloys occurs preferentially at low

angle GB [89]. In particular, the frequency of ATPs has been found to peak at 15%-20%

when the θ is smaller than 15° and to decrease steeply as θ increases, in such a way that,

when θ > 60°, the “crossing frequency” is smaller than 2%. Figure 5.8 illustrates the

variation in the crossing frequency with θ for the Mg AZ31 alloy sample compressed along

RD to a strain of 1.6%. Measurements were performed from 2D EBSD maps obtained

at the center of the specimen along the plane perpendicular to TD. In agreement with

Ref. [89], the crossing frequency peaks at boundaries with θ < 15° and becomes negligible

when θ > 50°.

It can be observed in Figure 5.5 that grain P1 is separated from neighboring grains P2,

P3 and P4 by three boundaries whose misorientation angle/axis pairs (θ/r) are: θP1−P2

= 15.7° [9, -14, 5, -13], θP1−P3 = 22.3° [2, -7, 5, -27], θP1−P4 = 64.3° [8, -22, 14, 7].

The three misorientation angles are pointed out in Figure 5.8 using black arrows. θP1−P2

(15.7°) is relatively small and belongs to the range where ATP form readily. θP1−P3

(22.3°) has an intermediate value included in the range where the formation of ATP starts

to become scarcer, and θP1−P4 (64.3°) is a high angle boundary that severely hinders

twin propagation [75, 76, 89]. In the following, the influence of the θ of the surrounding

boundaries on twin propagation is discussed. A note must be made here that, while

boundary misorientation also has a significant effect on slip activity, the latter will not be

studied here.

60

0 10 20 30 40 50 60 70 80 900

5

10

15

Misorientation angle (degrees)

Cro

ss

ing

fre

qu

en

cy

(%

)

Figure 5.8: Variation in the crossing frequency with respect to the GB misorientationangle measured in the sample compressed along RD up to a strain of 1.6%; the arrowspoint to the θ between P1 and P2, P1 and P3, and, P1 andP4.

5.3.2.1 Low misorientation angle boundary (θP1−P2= 15.7°)

The GB between grains P1 and P2 has a small misorientation, θP1−P2 = 15.7°. Moreover,

grain P2 is equally well oriented for tensile twinning. In fact, the active primary variant

T21 has a SF that is very similar to the one corresponding to the T11 variant. Thus, for

this particular case, this post-mortem study does not allow differentiation between the

following multiple scenarios for twin nucleation: twinning can take place first in grain P1,

in grain P2 or simultaneously in both grains. Figure 5.9 shows two 2D sections of the

volume of Figure 5.5 obtained at different heights in the vicinity of the P1 − P2 GB. It

reveals that T11 and T21 variants meet at the GB forming ATP at most of the heights

examined. These findings suggest that the low misorientation of the GB, together with

the fact that it separates two grains that are very well oriented for tensile twinning,

prevents the development of local stresses that may trigger non-Schmid plastic events.

The plasticity in the vicinity of this GB seems to fully comply with Schmid law and is,

thus, originated in response to the macroscopic applied stress.

5.3.2.2 Intermediate misorientation angle boundary (θP1−P3= 22.3°)

Grain P3 is less favorably oriented for tensile twinning than grain P1, and it is therefore

reasonable to assume that twinning first took place in the latter and that twin propagation

into grain P3 ensued. Figure 5.10 shows a sequence of 2D EBSD maps obtained from the

volume of Figure 5.5 at different heights in the vicinity of the P1 − P3 GB. It can be

seen that one of the T11 variants, labeled T111 in the figure, impinges at the boundary at

all the heights analyzed, triggering the nucleation of variant T32. At a height of 1.6 µm

(Figure 5.10(a)), variant T111 intersects GB P1−P3 at the triple point P1−P3−P4. Then

the intersection moves along the P1 − P3 boundary (Figures 5.10(b-o)), away from this

triple point. The SF of T32 is very low (SF = 0.0486) and, thus does not have a high


Figure 5.9: 2D EBSD maps in the RD of the GB between grains P1 and P2; map (a)corresponds to a height of 3.4 µm and map (b) to a height of 4.4 µm.

tendency to grow. Its thickness at the P1 − P3 GB is similar to that of the corresponding

T111 variant, but it thins down with the distance away from the GB, ending up in a pointed

tip. These findings illustrate how, as the boundary misorientation angle increases from

15.7° to 22.3°, the formation rate of twin variants deviating from Schmid law with respect

to the macroscopic stress state increases.

Furthermore, these data reveal that the 3D grain morphology and the boundary plane

may also contribute to alter the local stress state, and therefore play an important role in

variant selection. A second T11 variant, which is designated T112 in Figure 5.10, intersects

the triple point P1 − P3 − P4 and remains at that position for a relatively wide range of

heights (Figures 5.10(g-o)), leading to the nucleation of the primary variant T31 (SF =

0.2074). The selection of this variant type, and not of a T32 twin, as in the previous case,

appears to be deeply influenced by the relative orientation of the P3 − P4 GB and the

boundary of the twin generated. As shown in Figure 5.10(a), at the height at which the

T111 variant intersects the triple point, the P3 − P4 GB makes a ∼ 50° angle with the T32

variant generated. However, Figure 5.10(g) reveals that, at the height at which the T112

variant intersects the triple point, the P3−P4 boundary is parallel to the boundary of the

T31 transferred twin.

5.3.2.3 High misorientation angle boundary (θP1−P4= 64.3°)

The GB between grains P1 and P4 is highly misoriented, θP1−P4 = 64.3°. Figure 5.11 is a

sequence of 2D EBSD maps obtained from the volume of Figure 5.5 at different heights

62

Figure 5.10: Sequence of 2D EBSD maps of the GB between grains P1 and P3 in heightsteps of 200 nm; the first slice shown (a) corresponds to a height of 1.6 µm and the finalslice (o) to a height of 4.4 µm.


Figure 5.11: Sequence of 2D EBSD maps in the RD at height steps of 200 nm, whichillustrate the GB between grains P1 and P4; the first slice shown (a) corresponds to aheight of 1 µm and the final slice (h) to a height of 2.4 µm.

in the vicinity of the P1 −P4 boundary. It can be clearly seen how propagation of the T11

variants is severely hindered by the presence of the GB. This leads to the formation of high

local back-stresses, which trigger a series of non-Schmid effects in the original grain. In

particular, variant T12 is activated, and variant T112 undergoes “double” tensile twinning

(DTT) (pointed out in Figure 5.11 with a black arrow). Tensile twinning of a variant

T112 is forbidden under the sole effect of the applied stress (parallel to RD) and would

require the presence of an effective local stress that lies in a plane perpendicular to RD.

Figure 5.12 illustrates how DTT (shown by black arrows) as well as secondary and tertiary

variants also appear in the interior of the P1 grain. This effect is probably associated with

the build-up of high local stresses due to the presence of the boundaries of the primary

twins T11 [224].

In summary, the influence of the five parameters that characterize GB [225] on twinning

is still not well known, and this investigation analyzes the effect of one of them (θ). For

the first time, 3D aspects of twin transfer across GB have been investigated. The present

results confirm that the GB misorientation has a critical influence on twinning in the

Mg AZ31 alloy, as suggested recently [89]. More specifically, it was observed that low

misorientation GB between grains that are favorably oriented for twinning are regions

where plasticity can be explained by the Schmid law with respect to the macroscopic

applied stress. However, as the θ becomes larger, the probability of non-Schmid events,

such as the nucleation of low SF variants, increases. Finally, highly misoriented GBs

constitute strong obstacles to twin propagation and give rise to high back-stresses, which

trigger local non-Schmid plasticity in the original grain. This study reveals that 3D

64

Figure 5.12: 2D EBSD section of the volume illustrated in Figure 5.5, located at a heightof 4.8 µm.

microstructure characterization by the 3D EBSD technique is critical to relate individual

twinning events with microstructural parameters such as boundary misorientation. The

trends reported here were obtained from the observation of a single volume and, thus,

further work should be carried out to confirm the statistic validity of the conclusions drawn

here. However, these results are consistent with recent reports on the subject [48,75,76,89].

Rolled sheets and extruded bars of conventional Mg alloys usually have strong textures

and, when compressed along RD and the extrusion axis, respectively, many grains are

favorably oriented for tensile twinning [10]. However, depending on the processing

conditions, such as temperature, strain rate, die geometry or reduction ratio, such textures

will consist of single components with a high fraction of low misorientation GB or of fiber

textures with a large fraction of highly misoriented GB [57,78]. The present observations

support the suggestion by El-Kadiri et al. [75] that, in a polycrystal, the twinning stress

is sensitive to the distribution of GB misorientations (sometimes called GB texture).

Furthermore, this study illustrates that, for similar grain sizes, in materials with fiber

textures a high fraction of secondary and tertiary variants, as well as of DTT twins will

be activated, and, thus, a higher strain hardening is expected. The present findings

underline the complexity of twin-GB interactions and emphasize the need to incorporate

heterogeneities in the intragranular stress fields into numerical models [137,139,179,195].


5.3.3 Previous model vs. extended model

Figures 5.2 and 5.3 confirm that both the initial model described in Chapter 4 (where

each grain was represented by one unique element) and the extended one proposed in the

present work (where each grain is discretized by many elements) exhibit quasi-identical

averaged mesoscopic behavior. As stated in Section 5.2, this implies that the new model is

a priori equally calibrated to the experimental results of previous chapter. However, the

new model allows for the high resolution observation of complex intragrain mechanisms

not accessible to the initial model (and often experiments), and thus allows for the study

of complex phenomena such as the ones discussed in the next sub-section.

Another very important conclusion can be drawn from the observation of the quasi-

identical averaged mesoscopic behavior of both models. The study of HCP polycrystalline

aggregates at the mesoscale (i.e. where a thorough understanding of the intragrain

mechanics is not required) is adequately captured by a simple approach by modeling each

grain with one unique element. In other words, the stress-strain curve, texture evolution

and average slip and twin activities are accurately modeled by such a model. As the initial

model consists of only 2,592 elements vs. the 475,947 elements for the new model, the

choice of the former model can thus lead to significant savings in computing resources for

modeling studies focused mainly on mesoscopic mechanics.

5.3.4 Modeling of twin transfer and twin nucleation as a function of GB

misorientation angle

Twin transfers across the P1 − P2, P1 − P3 and P1 − P4 GBs were simulated using

the extended CPFE model described and validated in Section 5.2. To this end, a

polycrystalline model, consisting of a 25-grain Voronoı tessellation (a total of 2,113,409

tetrahedral elements) was used (Figure 5.13(a)). For each simulation, two grains in the

interior of the model, also shown in Figure 5.13(a), were assigned the orientations of the

corresponding P1−Pi |i=2,3,4 pair of grains, while the overall texture of the polycrystalline

aggregate was kept the same as in the experimental case. Figures 5.13(b-d) illustrate twin

propagation along the P1 − P2, P1 − P3 and P1 − P4 GBs, respectively (Movies 02-04

in Supplementary Material on CD-ROM). The active variants upon uniaxial compression

along RD are shown using different colors. For the P1 − P2 case (Figure 5.13(b)), the

primary variants, (1012)[1011] and (1012)[1011] are activated, as expected. Figure 5.13(c)

reveals that the primary twin variant (1012)[1011] is transferred across the P1 − P3 GB

to the secondary twin variant (1102)[1101] in grain P3. Finally, Figure 5.13(d) illustrates

the P1 − P4 case, where twin propagation is not occurring. Furthermore, simulations

also predict that twin propagation is easier across GB of small misorientation: in the

low angle GB (P1 − P2), it starts at a strain of 2.8%; in the moderately misoriented

GB (P1 − P3), it begins at a strain of 5%; in the high angle GB (P1 − P4), twinning is

not transferred (Movies 02-04 in Supplementary Material on CD-ROM). The predicted

cumulative slip/twin activity in the vicinity of the three GBs at these two strains has been

66

Figure 5.13: (a) Polycrystalline model (a quarter of the model is hidden to ease thevisualization of the two grains of interest) consisting of a 25-grains Voronoı tessellation(2,113,409 tetrahedral elements); the two grains of interest are assigned the orientationsof the corresponding P1 − Pi |i=2,3,4 pairs and cut transversally for visualization purposesof: (b) twin transfer across the P1−P2 boundary (3.9% of strain); (c) twin transfer acrossthe P1−P3 boundary (6% of strain); (d) twin transfer across the P1−P4 boundary (6% ofstrain); the red line shows the paths from grain P1 to the neighboring grains along whichslip activity was evaluated in Figure 5.14.

studied along the path colored in red in Figures 5.13(b-d) and is shown in Figure 5.14. At

the lowest strain level, little slip activity takes place in grains P1, P2 and P3. However, basal

slip is seen to be substantially active in grain P4, which is poorly oriented for twinning. At

the highest strain level, basal slip activity is seen to increase in the vicinity of the P1 −P2

GB, and both basal and non-basal slip are found to be active near the P1 − P3 GB and,

even more so, in the area close to the P1 − P4 GB.

The model is finally used for the prediction of twin nucleation in the low (P1−P2) and

high (P1−P4) misorientation GBs. Figure 5.15(a) shows the high resolution polycrystalline

model of grains P1 and P2 at a strain of 2.1% in which the “twin nucleation point” is seen to

occur at their GB (and more specifically, at a triple point). Figure 5.15(b), which depicts

the variation in the von Mises stress at the same strain along the path drawn in red in

Figure 5.15(a), indicates that a stress peak takes place at the GB, suggesting that twin

nucleation is triggered by the build-up of high local stresses. Additionally, Figure 5.15(c),

which corresponds to a strain of 2.5%, indicates that twin growth occurs simultaneously on

both sides of the GB and that basal slip activity peaks at the GB (Figure 5.15(d)). It must


Figure 5.14: Normalized slip/twin activities in the vicinity of the three GBs P1−P2, P1−P3

and P1−P4 at 2.8% and 5%, respectively, along the path drawn in red in Figures 5.13(b-d);the black dashed line indicates the GB.

68

be emphasized that the “nucleation point” indicates the rotation of the first element due

to twinning on the primary variant, but not the onset of twin activity. Homogeneous twin

activity on both sides of the GB very similar to that in Figure 5.15(d) was observed at

all lower strains (not shown here), thus indicating that the overall macroscopic stress

is also responsible for the generalized twinning observed in both favorably oriented

grains. In summary, the stress concentration at the low misorientation GB appears to

be the triggering mechanism early on in the deformation process and is quasi-immediately

accommodated by a Schmid-related homogenization of the twin activity on both sides

of the GB. This is consistent with earlier work by Proust et al. [141], which reported

that the CRSS for twin nucleation is higher than that corresponding to twin growth.

Figures 5.16(a) and (b) show the von Mises stress field and twin activity, respectively, of a

polycrystalline model of grains P1 and P4 (high misorientation) compressed up to a strain

of 0.7%. The activity of the primary variant in grain P1 peaks at a region in the grain

interior that is away from the P1 − P4 GB. This indicates, first, that twin nucleation is

not directly favored by such a highly misoriented GB and, second, that twinning activity

responds mostly to the macroscopic applied stress, and not to the local concentration of

von Mises stress (such as in the previous case).

These predictions are in full agreement with recent reports in Mg alloys and

titanium [48, 74, 75, 226–228], which highlight the preferred nucleation of twins in GB

with misorientations smaller than 30°. Furthermore, these results and in particular

Figure 5.15(b) are also consistent with recent atomistic simulations, which demonstrate

nucleation of tensile twins at GB via a pure-shuffle mechanism originated by a high

local stress concentration. Finally, as mentioned earlier, experimental 2D or 3D EBSD

observations do not allow one to differentiate whether ATP at low angle boundaries

formed by simultaneous nucleation of both twins forming the pair, or whether, on the

contrary, nucleation took place first in one grain and twin transfer ensued. As such, the

extended modeling approach provides a new numerical tool able to complement 3D EBSD

in determining twin dynamics at the microscale.

5.4 Conclusions

In this chapter, a combined 3D EBSD and modeling approach was adopted to investigate

the effect of GB misorientation on tensile twin propagation in a rolled Mg AZ31 alloy. For

the first time, 3D aspects of twin transfer across GBs have been investigated. The material

was compressed along RD to small strains in order to nucleate a large number of twins,

but prevent significant twin growth. 3D EBSD was then carried out in a selected volume

in which a central grain, very well oriented for tensile twinning, is surrounded by three

GBs with a wide range of θ. Twin transfer across these boundaries was also simulated

using an extended CPFE model able to account for the topological information of the

grains. The following conclusions may be drawn from the present study:

5.4. Conclusions 69

Figure 5.15: (a) Polycrystalline model cut transversally for visualization purposes showingthe twin nucleation point at the low angle P1 −P2 GB and (b) variation of the von Misesstress along the path drawn in red in (a) at a strain of 2.1%; (c) polycrystalline model and(d) normalized slip/twin activity along the path drawn in red in (c) at a strain of 2.5%).

Figure 5.16: Polycrystalline model cut transversally for visualization purposes at a strainof 0.7% showing grains P1 and P4 with the fields of (a) the von Mises stress and (b)the activity of the primary variant in grain P1 (the black arrow indicates the point ofnucleation).

70

1. The 3D morphology of twin variants appears to be related to their SF. Primary

variants with high SF tend to have plate-like morphologies, with approximately constant

thickness, and encompass the entire parent grain. However, secondary and tertiary

variants, with SFs smaller than ∼ 0.2, have a more irregular geometry, which is related to

the heterogeneous nature of the local effective stress field.

2. The GB misorientation angle has a critical influence on twin propagation. GBs

with θ < 15° facilitate the formation of ATP, and plasticity in the vicinity of these GBs

is driven by the macroscopic applied stress. When the θ becomes larger, the probability

of non-Schmid events, such as the nucleation in neighboring grains of low SF variants,

increases. Finally, highly misoriented GB constitute strong obstacles to twin propagation

and give rise to high back-stresses, which trigger local non-Schmid plasticity in the original

grain.

3. Variant selection in the transferred twins appears also to be critically influenced by

the 3D grain morphology, which, in turn, contributes to alter the local stress state.

4. The proposed extended CPFE model successfully predicts that twin transfer is easier

in GB with low θ and that it is severely hindered in highly misoriented GB. Moreover, the

model is able to capture the active variants in both the original and neighboring grains.

The predictions of the model are, thus, in full agreement with the experimental results

described above.

5. The model corroborates the build-up of increasingly high local stresses at twin-

GB interfaces with increasing θ and, furthermore, it predicts that the presence of such

stresses gives rise to increasingly higher activity levels of non-basal slip in the vicinity of

the corresponding boundaries.

6. The model predicts that twin nucleation is favored at GBs with low misorientation

as a consequence of the high local stress concentrations, but that propagation is

accommodated by the overall macroscopic stress. However, highly misoriented boundaries

are not preferential sites for twin nucleation, despite having a high level of von Mises

stress.

7. The model failed to predict the non-Schmid effects (DTT, secondary and tertiary

twin variants) observed when the GB presents a high misorientation angle. According to

some studies [75,128,229–231], non-Schmid effects are due to a special case of back-stresses

which are not arising from geometrically necessary dislocations (GNDs) but from the GB-

twin interactions [95]. Nevertheless, GNDs have been shown to play a role in the evolution

of back-stress at GBs [232, 233]. Both observations might be reconciled by hypothesizing

that GNDs do not trigger the nucleation of the non-Schmid effects but participate in

their development. Therefore, further studies are required to develop crystal plasticity

models that can contemplate the complexity of the mechanisms derived from local stress

fluctuations in the vicinity of the GB and in the interior of the grains.

CHAPTER 6

Effect of hydrostatic pressure on the 3D porosity distribution and on the

mechanical behavior of a high pressure die-cast Mg AZ91 alloy

An approach to reduce the detrimental effect of porosity in a HPDC Mg AZ91 alloy

consisting on applying a hydrostatic pressure after HPDC processing is investigated. The

effect of the pressure treatment on the 3D porosity distribution and on the mechanical

behavior of the samples is studied using an experimental-computational methodology. The

3D distribution of pores before and after the pressure treatment is analyzed by XCT. A

relatively large fraction of pores reconstructed from XCT experimental measurements is

used as an input for a FE model. This model is ultimately utilized to simulate and analyze

the effect of pressure on the 3D pore distribution as well as on the mechanical response of

the material.


6.1.1 Material and process

A Mg AZ91 alloy was HPDC processed into a dog-bone-shaped die. The geometry and

dimensions of the castings are shown in Figure 6.1. Since, as it will be seen later, the

porosity distribution is highly dependent on the location in the cast specimen, four

representative regions were identified and labeled in Figure 6.1 as “Grip 1”, “Grip 2”,

“GL1” (the region of the gage length closer to Grip 1) and “GL2” (the region of the

gage length closer to Grip 2). Multiple specimens were prepared using the following

processing conditions. The molten material was injected at a liquid alloy temperature

of 630-670 � and a gate velocity of 8-9 m/s, using a servo-hydraulic HPDC computer-

controlled machine. The injection pressure was 400-500 bar and the mold temperature

was set in the range of 190-230 �.

71

72

Figure 6.1: Sample geometry of the Mg AZ91 alloy dog-bone shaped casting fabricated byHPDC.

Figure 6.2: Microstructure of the as-HPDC Mg AZ91 alloy at different magnifications; themicrographs were taken in Grip 1.

The microstructure of the as-received material was analyzed by OM in an Olympus

BX51 optical microscope. The microstructure is consistent with previous observations of

HPDC Mg AZ91 alloys [234]. It is formed by α-Mg “islands” (light gray) surrounded by

β-phase, Mg17Al12 particles (dark gray). Figure 6.2 includes several optical micrographs

illustrating the microstructure at different magnifications. Figures 6.2(a-b) reveal that

the β-phase is homogeneously distributed throughout the microstructure. Both gas and

shrinkage pores are present (selected examples are labeled in Figure 6.2). The former have

a large size variation ranging from a few micrometers up to 200 µm in diameter. Shrinkage

pores were found isolated as well as emanating from the gas pores. In Figure 6.2(c), some

grains are visible in the vicinity of a pore. The average size of such grains is ∼ 10 µm. It

can be seen that grains are in general smaller than the dendrite width.

A pressure treatment of 600 MPa at room temperature during 10 min was first

applied to a 1.8 mm diameter and 6 mm long cylinder extracted from Grip 1, using a

Hiperbaric 420 machine from Hiperbaric S.A. A disk of 1.8 mm in diameter and 820 µm

in thickness corresponding to the central part of this cylinder was examined by XCT (as

6.1. Experimental procedure 73

described in more detail in the next sub-section) before and after pressurization in order

to evaluate the effect of pressure on the evolution of the porosity distribution. The same

pressure treatment was then applied to several HPDC dog-bone-shaped castings with the

aim of investigating the effect of pressure on the mechanical behavior before and after

pressurization.

6.1.2 X-ray computed tomography inspection

Quantitative information of the 3D porosity distribution was obtained by means of XCT

using a Nanotom 160NF tomograph from Phoenix. All the tomographic measurements

were performed utilizing a tungsten target. Five as-HPDC dog-bone-shaped specimens

were first inspected at low resolution to identify qualitatively the different porosity levels

throughout the samples. The conditions for the collected low resolution tomograms were

130 kV, 90 µA, 1,000 projections and 500 ms exposure time. The pixel size was set to

21.7 µm. A second measurement at a higher resolution of 4.9 µm pixel size, 90 kV, 120 µA,

2,000 projections and 500 ms exposure time was performed on different parts of the samples

to better identify porosity. Finally, in order to obtain the highest possible resolution, the

cylinder of 1.8 mm in diameter and 6 mm in length, described in the previous sub-section,

was machined out of the high porosity region (Grip 1) and the central part of this cylinder

(a disk of 1.8 mm in diameter and 820 µm in length) was measured with a pixel size of

1 µm, 90 KV, 100 µA, 2,000 projections and 750 ms exposure time. This cylinder was

then subjected to a hydrostatic pressure of 600 MPa at room temperature during 10 min

and the same disk was measured again under the same conditions by XCT in order to

evaluate the influence of the applied pressure on each single pore. All the tomograms were

reconstructed using an algorithm based on the FBP procedure for Feldkamp cone beam

geometry [235].

6.1.3 Image processing and void tracking

Identification of the cavities was carried out by selection of the voxels as belonging either to

a void or to the bulk matrix material based on their gray level. Prior to the segmentation

of cavities, all the slices forming the volume were equalized to have a homogeneous gray

level of the matrix along the volume. The cavity segmentation procedure used was

based on the local variance method from Niblack applied to each slice by adapting the

threshold according to the mean and standard deviation of the peak belonging to the

matrix material [236]. For the two reconstructed samples (before and after the hydrostatic

pressure treatment) only cavities having a volume larger than 200 voxels (equivalent radius

of ∼ 3.6 µm) were selected after segmentation for analysis. This lower limit was chosen

according to Ref. [237], which showed that an ellipsoid can be well approximated by

structure larger than 5×5×5=125 voxels. The complexity factor (CF) parameter [238]

was used to characterize the cavity shape. As it is shown in Figure 6.3(a), the CF is

74

related to the exclusion volume resulting from the intersection of the real cavity with its

equivalent ellipsoid (same volume and moment of inertia). The CF can be calculated as:

CF =V1 + V3

VC(6.1)

where VC = V1 + V2 is the volume of the cavity, composed of the volume of the cavity

outside the equivalent ellipsoid (V1) and the volume of the cavity inside the ellipsoid (V2),

and V3 is the volume of the ellipsoid which is not occupied by the cavity (Figure 6.3(b)).

(a) (b)

Figure 6.3: (a) Schematic representation of CF parameter: a cavity (in blue) isapproximated by its equivalent ellipsoid (in red) and a, b and c are the axes of theequivalent ellipsoid; (b) CF parameter definition [201].

The effect of pressure on the volume and shape of the pores was analyzed by tracking

each individual pore in the disk before and after pressurization. Prior to void tracking, the

two volumes of the disk, measured with the highest resolution, were translated and rotated

using the Image Fusion module of MedINRIA [239] for image registration. The registration

was performed using manual landmarks and rigid body options to perform translation and

rotation, but not scaling. The cavities found in the initial volume were tracked in the

pressure, i.e. treated volume using an in-house image-registration algorithm developed in

Matlab and based on the phase-correlation method centered on each cavity [240,241]. For

that purpose, an area of interest (AOI) centered on the maximum area of the selected

cavity was chosen. The AOI size was sufficiently large to contain a few extra features

(neighbor cavities or highly absorbing particles) to improve the correlation coefficient in

the local area. By using a proximity criterion, based on a minimum distance from the

center of gravity of the initial cavity to any pixel of the pressure-treated cavity, the cavities

were considered as found or lost.

After the pressure treatment, gas pores linked by a crack or a shrinkage pore may in

some cases lose such link as a result of closure of the linking crack or shrinkage pore. In

this case a single cavity in the initial state splits into two or more cavities in the final

state. This effect was considered by detecting the number of such final state cavities,

6.1. Experimental procedure 75

called “predecessor” cavities [240], and was taken into account in the evaluation of cavity

evolution by removing them from the analysis.

6.1.4 Mechanical testing

In order to study the effect of the applied pressure on the mechanical behavior of the

HPDC Mg AZ91 alloy, twelve cylindrical specimens of about 3 mm in diameter by 4.5 mm

in length were extracted from a high porosity region (Grip 1) of the HPDC dog-bone

specimens (six from non-treated samples and six from pressurized specimens) and were

tested under uniaxial compression at room temperature in an electromechanical universal

testing machine (Instron 3384) at constant strain rate of 10−4 s−1. The load was monitored

with a 10 kN cell.

6.1.5 Finite element analysis

The influence of the applied pressure on the 3D porosity distribution and on the mechanical

behavior of the HPDC Mg AZ91 alloy was analyzed by FE. The FE mesh was built from

a 300×300×300 µm3 (1 pixel=1 µm2) tomographic sub-volume extracted from the high

resolution XCT measurement of the 1.8 mm diameter and 820 µm-thick disk (Figure 6.4).

This sub-volume contained ∼ 200 complete pores (i.e. not in contact with the sub-volume

faces). The FE mesh was built as follows. A surface mesh was created in StereoLithography

(STL) format from the tomographic binarised data using VGStudio Max software [242],

(a similar methodology was used in Ref. [243] for cellular materials). The number of

surface elements was set to achieve an optimal resolution in order to capture the shape of

the pores as accurately as possible. The STL surface-meshed volume was then imported

into Hypermesh [244] and meshed with 792,681 hybrid quadratic tetrahedral elements

(1,129,194 nodes). In order to reduce the number of elements the finest mesh in the

neighborhood of the pores was gradually coarsened towards pore-free regions. The meshed

volume was finally exported to the FE program ABAQUS/Standard [245] in order to carry

out the simulations. The behavior of the bulk material was described using standard J2

plasticity with linear isotropic hardening. Note that this implies that the crystallographic

microstructure (grain size, texture, etc.) is not accounted for (discussion in Section 6.3).

Accordingly, the von Mises yield stress σy is defined as:

σy(ǫpl) = σ0 +Hǫpl (6.2)

where σ0 is the initial yield strength, ǫpl is the equivalent plastic strain and H is hardening

modulus. A Young’s modulus of 44.8 GPa and a Poisson’s ratio of 0.35 were used as

the elastic properties of the Mg AZ91 alloy [246]. The initial yield stress σ0 and the

hardening parameterH were obtained from uniaxial compression tests of cylinders of 4 mm

in diameter and 6 mm in length machined out from a very low porosity region (GL2). The

average porosity in GL2 was smaller than 1%. Previous studies [247] have shown that

these parameters are insensitive to porosity below such levels, thus suggesting that the

76

Figure 6.4: FE mesh of the selected tomographic sub-volume geometry: a sector of thepore-free mesh was removed for visualization purposes.

proposed approximation is reasonable. The tests were done with an electromechanical

universal testing machine (Instron 3384) at constant strain rate of 10−4 s−1 and the load

was monitored with a 10 kN load cell. Figure 6.5 shows the experimental and simulation

results for σ0=160 MPa and H=2.2 GPa.

The final model, i.e. using the mesh shown in Figure 6.4 and the constitutive model

calibrated in Figure 6.5, was subjected statically to a hydrostatic pressure of 600 MPa,

followed by relaxation. The effect of pressure on the 200 complete pores was tracked. The

mechanical response of the sub-volume before and after the simulated pressure treatment

was ultimately analyzed by simulating a uniaxial compression test. In both cases the two

compressed faces were not laterally constrained.

6.2 Results

6.2.1 Experimental results

6.2.1.1 Qualitative analysis

Figure 6.6 illustrates the low resolution tomographic reconstruction of the complete dog-

bone initial sample, the high resolution reconstruction of the disk from the cylinder

subjected to the high pressure treatment (extracted from Grip 1) before the application

of the pressure, and the sub-volume of 300×300×300 µm3 used in the FE model. The low

resolution reconstruction exhibits different void sizes and concentrations along the sample

axis. In particular, larger voids and a higher void density can be found in the grips.

Grip 2 contains mostly globular pores while in Grip 1 a large concentration of cracks and

shrinkage pores can be observed. The gauge length also presents a heterogeneous spatial

distribution of pores. In particular, the pore density is higher in the region close to Grip

1 (Figure 6.6). The void volume fraction varies in different regions of the sample from

6.2. Results 77

0.00 0.03 0.06 0.09 0.12 0.15 0.180

50

100

150

200

250

300

350

400

450

True

str

ess

(MPa

)

True strain

experiment GL2 simulation - model calibration

Figure 6.5: Uniaxial compression true stress-true strain curves: experimental data andmodel calibration.

an average value of about 9% (range: 5% to 15%) in the grips to 0.7% (range: 0.5% to

1.0%) in the gauge length. Even in the smaller disk volume measured by high resolution

tomography the void volume fraction varies in the untreated specimen from 5% to 13.4%.

Preliminary 2D inspection of the reconstructed tomographic slices before and after the

application of the pressure exhibits a reduction in the number of pores, in the pore size, as

well as in the occurrence of changes in the pore geometries. As an example, Figures 6.7(a-

b) show two reconstructed slices, before and after the pressure treatment, respectively, of

the same cross-section of the HPDC specimen after a correlation procedure. Two areas

within these cross-sections have been enlarged and are shown in Figures 6.7(c-d). Most of

the pores shown in Figure 6.6(d) are smaller than their counterparts prior to treatment

Figure 6.7(c) and their shapes have also been altered. Finally, it can be noted that the

thin wall separating the central large pore from its neighbor (indicated by the arrows)

undergoes significant plastic strain and is wider after the pressure treatment. This effect

was frequently observed in the inspected volume.

The 3D effect of the pressure treatment on the pores can be better appreciated in

Figures 6.8(a-c). The sub-volume extracted for the FE model was selected as a RVE of

the overall disk. The size of this RVE was chosen as a balance between representativeness

and visualization, since the large amount of pores contained in the disk would hinder the

3D visualization. The porosity distribution before and after pressure treatment can be seen

in Figures 6.8(a-b), respectively. The reduction in pore volume is visually noticeable in the

majority of the pores by superposition of both volumes in Figure 6.8(c). A closer inspection

of Figures 6.7 and 6.8 shows that the hydrostatic pressure affects more strongly the cracks

or shrinkage pores arising from the surface of the globular gas pores, by closing them

78

Figure 6.6: Low resolution tomographic reconstruction of the complete dog-bone casting,high resolution reconstruction (initial state) of the cylindrical volume subjected to thepressure treatment and of the sub-volume of 300×300×300 µm3 used in the FE analysis.

Figure 6.7: Reconstructed cross-sectional views of the same slice before (a) and after (b)the pressure treatment; magnified region at the largest area of the central pore before (c)and after (d) the pressure treatment.

6.2. Results 79

Figure 6.8: 3D reconstruction of the sub-volume extracted for simulation purposes before(a) and after (b) the pressure treatment; (c) superposition of volumes from (a) and (b)with initial pores (before treatment) set to semitransparent.

completely, within the limits of the resolution used in the XCT studies (some are indicated

by arrows in Figure 6.8). This is expected as high stresses normally concentrate at the

acute angles formed by the cracks around the pores, causing a larger volume reduction

and shape alteration.

6.2.1.2 Quantitative analysis of pore volume and shape change

The combination of XCT with powerful image analysis techniques allows for a statistical

evaluation of the evolution of a large number of cavities subjected to the pressure

treatment, using the procedure explained in Sub-section 6.1.3. Only cavities composed

of more than 200 voxels were considered in the analysis. A total of 17,649 cavities larger

than 8 voxels were found in the initial volume, while 13,549 cavities remained after the

treatment. From the latter set, it was possible to track 9,689 cavities larger than 200 voxels

which fulfilled the proximity criterion and only had one predecessor. The volume change

of the initial cavities which split in more than one predecessor cavity can be obtained by

comparing the sum of volumes of the predecessor’s cavities to the initial one. Only 2,164

cavities had split in more than one predecessor cavity, and were not considered in this

analysis for simplicity.

80

Figure 6.9(a) shows the histogram of the relative volume change ∆V/V0 for the 9,689

tracked pores, where ∆V is the difference between the initial volume V0 and the final

volume V. The distribution is bimodal showing a first peak at around 6%, and a second one

at 37%, the latter being fairly symmetric and relatively wide. The first peak is composed

of 1,742 cavities, while 7,947 cavities belong to the second one, i.e. 78% of the cavities have

experienced an average volume change of 37%. This arguably counterintuitive bimodal

distribution is thought to be a reasonable indication that some pores (belonging to the

first peak) contain pressurized gas arising from the HPDC process, thus preventing them

from further shrinking. The average volume change of the whole cavity population is 33%.

In an attempt to better understand the effect of pressure on the pore volume change, the

pore volume after the pressure treatment vs. the relative volume change is plotted in

Figure 6.9(b). It can be seen that volume changes below 15% are associated to a wide

range of pore sizes, suggesting that pores with internal pressure have a large variety of

sizes. Volume changes larger than 45% correspond mainly to pores whose volume is less

than ∼ 3,500 µm3. Additionally, the pores with volumes larger than 5,000 µm3 (belonging

to the second peak) undergo an average change of 33% (Figure 6.9(b)) while smaller pores

undergo changes within the whole range.

The ellipsoid fitting procedure for pores allows for the evaluation of pore shape. This

study is focused on the CF [238] described above. Figure 6.10(a) is a histogram of the

relative CF change ∆CF/CF0, where ∆CF is the difference between the initial CF0

and the final CF. The maximum of the relative CF change is centered at ∼ 0% but

the ∆CF/CF0 distribution is slightly asymmetric towards negative values. A negative

change in ∆CF/CF0 indicates that the pores become more irregular after the pressure

treatment. The variation of ∆CF/CF0 with ∆V/V0 is shown in Figure 6.10(b). It

can be observed that the pores that belong to the first peak of relative volume change

(∆V/V0 ∼ 6%) tend to have positive values of ∆CF/CF0, i.e. they tend to become more

regular after pressurization. The internal pressure of these pores seems to influence their

plastic deformation during pressurization, making them more regular. ∆CF/CF0 tends

to subsequently decrease from positive values to negative ones (more irregular pores after

the pressure treatment) as ∆V/V0 increases. For pores for which ∆V/V0 is close to 27%,

the average ∆CF/CF0 is close to 0%. As ∆V/V0 becomes larger than 40% (with pores

whose volume is less than ∼ 3,500 µm3, Figure 6.9(b)), the average ∆CF/CF0 adopts

increasingly more negative values.

6.2.2 Simulations results

FE simulations were performed on the numerically extracted sub-volume from Grip

1 (Figures 6.4 and 6.6) following the procedure explained in Sub-section 6.1.5. The

experimental results show a bimodal distribution of the volume change consistent with

internal pressure on some of the pores formed during casting process. The pores with a

volume change smaller than 15% were assumed to have internal pressure and 7 pores

6.2. Results 81

0 10 20 30 40 50 60 700

50

100

150

200

250

300

Cou

nts

V/V0 (%)(a)

0 10 20 30 40 50 60 70102

103

104

105

106

Pore

vol

ume

afte

r tre

atm

ent (

m3 )

V/V0 (%)

(b)

Figure 6.9: (a) Histogram of the relative volume change (∆V/V0) of the 9,689 trackedcavities. (b) Pore volume after pressure treatment vs. relative volume change.

82

-60 -40 -20 0 20 400

255075

100125150175200225250

Cou

nts

CF/CF0 (%)

(a)

0 10 20 30 40 50 60 70-100

-75

-50

-25

0

25

50

75

100

CF/CF 0 (%

)

V/V0 (%)(b)

Figure 6.10: (a) Histogram of the relative CF change (∆CF/CF0) of the 9,689 trackedcavities; (b) relative CF change (∆CF/CF0) vs. relative volume change (∆V/V0) (positivevalues of ∆CF/CF0 indicate transition to more regular pores, while negative valuesindicate transition to more irregular pores).

6.2. Results 83

fulfilling this condition (out of ∼ 200) were identified in the simulation sub-volume

counterpart. Two simulations were then carried out, with and without internal pressure

in the 7 selected pores. The internal pressure in the pores was mimicked by defining a

quasi-incompressible behavior of the pores (but with a negligible shear modulus), with

a bulk modulus calibrated so as to capture the first peak of the relative volume change

distribution at ∼ 6%.

Figure 6.11 compares the experimental relative volume change data corresponding to

the pores included in the sub-volume used for simulations with the predictions of the

model with and without internal pressure in the selected pores. The average experimental

relative volume change is ∼ 37% which is consistent with the average relative volume

change experienced by the larger disk-shaped sub-volume that was previously analyzed

(Figure 6.9). The peak observed at small ∆V/V0 (< 15%) in Figure 6.9 is also apparent

in Figure 6.11, albeit with a significantly smaller height, since it only includes 7 pores

and is lacking statistical significance, (this is equivalent to ∼ 3.5% of the total number

of pores in the simulation sub-volume while up to 22% of the pores belong to this group

in the larger experimentally analyzed volume). The distribution of the simulated ∆V/V0

without considering the internal pore pressure consists of a single peak and the average

∆V/V0 is ∼ 23%. Taking into account the pores with internal pressure gives rise to a

bimodal distribution in which the peak at low ∆V/V0 includes the same 7 pores mentioned

earlier (Figure 6.11). The average ∆V/V0 is 23.5% for the simulation with internal pore

pressure which is still below the 37% measured experimentally in the same sub-volume.

Comparison between the pore volume after pressure treatment vs. the relative volume

change in Figure 6.11(b) confirms that the simulations succeed in better capturing the

behavior of pores larger than 5,000 µm3, that undergo a volume change of around 33%,

while it fails at capturing the tail of pores that undergo larger volume changes, that

correspond to pores whose volume is less than ∼ 3,500 µm3. The possible reasons for this

discrepancy will be discussed below.

Figure 6.12 compares the distributions of the relative change in the CF (∆CF/CF0)

corresponding to the pores included in the simulation sub-volume, for the experiment

and the simulation with and without internal pressure. The average value of ∆CF/CF0

obtained experimentally is -8%, while both simulations render a value of about -13%

(Figure 6.12(a)). As opposed to the previous study, simulations capture adequately the

average value of CF. Both experiments and simulations show that the ∆CF/CF0 decreases

as ∆V/V0 increases, in agreement with the trend illustrated in Figure 6.10(b). The pores

for which ∆V/V0 < 15% become slightly more regular after pressurization. This effect

is also captured by the simulations performed with internal pressure in the selected pores

yielding a value close to zero. The pores undergoing a larger volume change tend to

adopt a more irregular geometry after the pressure treatment. Again and as it will be

discussed later, the tail of the ∆CF/CF0 vs. ∆V/V0 distribution (Figure 6.11(b)), formed

by the pores which undergo a relative volume change larger than 40%, i.e. those for which

∆CF/CF0 decreases more, is not captured by the model.

84

0 10 20 30 40 50 60 70 800.0

0.2

0.4

0.6

0.8

1.0

Nor

mal

ized

num

ber o

f por

es

V/V0 (%)

Sub-volume for simulation Experiment Simulation without

internal pressure Simulation with

internal pressure

(a)

0 10 20 30 40 50 60 70 80102

103

104



internal pressure

Pore

vol

ume

afte

r tre

atm

ent (

m3 )

V/V0 (%)

(b)

Figure 6.11: Comparison between experimental and simulations results with and withoutinternal pressure for the same sub-volume; (a) normalized number of pores vs. the relativevolume change (∆V/V0); (b) pore volume after pressure treatment vs. relative volumechange (∆V/V0).

6.2. Results 85

-60 -30 0 30 600.0

0.2

0.4

0.6

0.8

1.0N

orm

aliz

ed n

umbe

r of p

ores

CF/CF0 (%)



internal pressure

(a)

0 10 20 30 40 50 60 70 80-100

-75

-50

-25

0

25

50

75

100 Sub-volume for simulation Experiment Simulation without internal pressure Simulation with internal pressure

CF/CF 0 (%

)

V/V0 (%)

(b)

Figure 6.12: Comparison of experimental results and simulations with and without internalpressure for the same sub-volume; (a) normalized number of pores vs. the relative CFchange (∆CF/CF0); (b) relative CF change (∆CF/CF0) vs. relative volume change(∆V/V0)

86

6.2.3 Mechanical tests

Uniaxial compression tests were performed on both the as-received material and the

pressure treated material. The corresponding true stress-true strain curves are shown

in Figure 6.13(a). The pressure treatment results in an increase of the stress over the

whole plastic region, with an increment of 31.5 MPa. Similar strain hardening behaviors

are observed before and after pressurization.

The influence of the pressure treatment on the mechanical behavior was also

investigated with the FE model. The results of the simulation, shown in Figure 6.13(b),

also reveal that the pressure treatment induces an improvement of 26.4 MPa of the

flow stress over the plastic region, in close agreement with the experimental results.

However, the simulations predict higher absolute flow stress values than the experimental

measurements. This is undoubtedly a result of the particular pore volume fraction

contained in the simulated sub-volume (Figure 6.6). While in the simulated sub-volume

the volume fraction of pores was about 5%, the real pore volume fraction reached local

values of up to 13.4% in the experimentally compressed cylinders. Additionally, pores

smaller than 200 pixels were not simulated. This naturally leads to higher simulated flow

stresses but, despite these differences, the model was able to capture the beneficial effect

of the pressure treatment quantitatively.

6.3 Discussion

The evolution of the porosity distribution of an HPDC Mg AZ91 alloy upon pressurization

has been analyzed by XCT and simulated in a sub-volume of the material by FE analysis,

using the experimentally determined distribution of pores in this particular sub-volume

and J2 plasticity with linear isotropic hardening for the pore-free matrix. In general,

a good agreement is found between experiments and simulations regarding the change

in volume and in geometry of a large number of individual pores. The study reveals

that a large fraction of pores (around 22%) is relatively insensitive to the pressurization

treatment, presumably due the presence of internal pressure as a result of the HDPC

process. The study also reveals a good agreement between experiments and simulations for

pores larger than 5,000 µm3, that undergo a relative volume reduction of ∼ 33%. However,

XCT reveals that a significant number of pores with volumes after pressurization smaller

than 3,500 µm3 undergo relative volume changes between 45% and 65% and these are not

captured by the simulations. These same pores experience large reductions in their relative

CFs (Figure 6.12(b)), i.e. they adopt more irregular shapes after the pressure treatment.

An aspect that might play an important role in this discrepancy is the plastic anisotropy

inherent to the HCP Mg lattice [78], which is not taken into account in the model. Pores

with final volumes between 200 µm3 and 3,500 µm3 undergoing relative volume changes

between 45% and 65% have initial volumes comprised between approximately 350 µm3

and 6,400 µm3, i.e. with equivalent radii between 4.4 µm and 11.4 µm. A close inspection

6.3. Discussion 87

0.00 0.02 0.04 0.06 0.080

50

100

150

200

250

300

True

str

ess

(MPa

)

True strain

Before pressure treatment - experimental After pressure treatment - experimental

(a)

0.00 0.02 0.04 0.06 0.080

50

100

150

200

250

300

True

str

ess

(MPa

)

True strain

Before pressure treatment - simulation After pressure treatment - simulation

(b)

Figure 6.13: Uniaxial compression response of (a) cylinders extracted from Grip 1 beforeand after the pressure treatment as measured experimentally and (b) the simulated sub-volume.

88

of Figure 6.2(c) reveals that the grain size ranges from approximately 5 µm to 10 µm.

Thus, the size of the pores that show the discrepancy is of the same order of magnitude as

the grain size and are, therefore, surrounded by a limited number of grains, not exceeding

about 10. These grains, depending on their orientation, will undergo different amounts of

plastic strain due to the inherently anisotropy of plastic deformation and the high local

shear stresses built-up around the pores. This can lead to larger pore volume reductions

and to more irregular pore geometries for pores with sizes in the range of the grain size.

This effect is expected to be strong in Mg alloys because the CRSSs of the different

slip and twin systems in Mg alloys at room temperature might span up to 2 orders of

magnitude [33].

This thorough study also brings the opportunity to analyze the effect of neighboring

pores on the evolution of the porosity during the pressure treatment. Figure 6.14 shows

the equivalent plastic strain field in four different cross-sections of the simulated sub-

volume after the hydrostatic pressure was released. As expected, the material at the pore

surfaces undergo large plastic strains and, interestingly, the strain fields of neighboring

pores interact with each other, provided that they are close enough. Figure 6.14 also

reveals that the extent of the plastic strain field around a given pore increases with its

size. According to the literature on void growth under hydrostatic stresses [248], pore

interactions are no longer negligible when the surface to surface distance is smaller than

the pore diameter, which is consistent with the simulations. Moreover, the experimental

evidence presented in Figures 6.7(c-d) reveals that, in the vicinity of large pores, significant

plastic strain occurs in the thin ligaments that separate these pores from their neighbors,

suggesting that those pores that have large neighboring pores should undergo larger pore

size alteration under pressure.

6.3.1 Ligament ratio analysis

In order to test whether these experimental observations have any statistical significance,

the extensive pore population studied here, with pore sizes ranging from 200 µm3 to

3,000,000 µm3 was analyzed using an approach based on nearest neighbor distance.

In order to account for the size of the neighboring pore, the ligament ratio, Rw =

w/(w + rNN ), was computed for each pore, where w is the surface to surface (ligament)

distance and rNN is the radius of the corresponding neighboring pore. This parameter

varies between 0 for a very small wall distance or a very large neighbor pore and 1 for a

large ligament distance or a very small pore. The smaller this parameter is for a given pore,

the higher the effect the neighboring pores should have on its size reduction. For instance,

a small neighboring pore with a very short ligament distance is expected to affect the pore

less than a large neighboring pore whose ligament distance is slightly larger, and this is

reflected in a lower value of the Rw for the latter case. Only the first 6 neighbor pores

with the minimum Rw were considered, assuming that they have a high probability of

being first order nearest neighbors and that they are spatially homogeneously distributed.

6.3. Discussion 89

Figure 6.14: Examples of the equivalent plastic strain field located around the pores afterreleasing the hydrostatic pressure.

Figure 6.15 shows the result of plotting ∆V/V0 as a function of Rw. The different curves

correspond to different initial volume classes (ranges of pore volumes in logarithmic scale).

For values of the Rw between 0.1 and 0.25 (i.e. when neighboring pores are large compared

to the ligament width) all the volume classes experience a large pore size reduction ∆V/V0

of about 44%, indicating a strong interaction between the strain field of the pore under

analysis with the ones of its neighbors. For values of the Rw higher than 0.25, i.e. as the

ligament width increases with respect to the size of the neighboring pore, two different

trends are observed as a function of initial pore size. For pores with volumes larger

than 1,030 µm3, the pore size reduction ∆V/V0 decreases as Rw increases, pointing to a

smaller effect of neighboring pores. For pores with volumes up to 1,030 µm3, the pore size

reduction, ∆V/V0, remains high and decreases with Rw only slightly. This is consistent

with the fact that these pores correspond to pores with sizes in the range of the grain size,

and therefore the local microstructure has an important effect on the pore size reduction

of these small pores. As such, for Rw values of about 0.8 the ∆V/V0 of small pores reaches

values close to twice the ∆V/V0 for large pores, in agreement with Figure 6.9(b).

As a consequence, this analysis reveals that the presence of neighboring pores influences

significantly the pore size reduction upon pressurization. The results are in consistent

with FE analysis performed by Tvergaard [249] on the interaction of pores of different

sizes under high stress triaxialities, showing that smaller voids grow more rapidly than

larger voids if the void volume fraction is sufficiently high (above ∼ 0.5%). It must

90

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.920

25

30

35

40

45

50

V/V 0

(%)

Rw = w/(w+rNN )

Volume classes ( m3) 200-454 455-1,030 1,031-2,338 2,339-5,307 5,308-12,046 12,047-27,339

Figure 6.15: Relative volume pore change (∆V/V0) vs. Rw = w/(w + rNN ) for severalinitial pore volume classes.

aslo be emphasized that the method presented in this work could also be used to study

experimentally the effect of different void volume fractions on the relative volume change

of voids of various sizes, specially below a radius of 3 µm, provided that a higher spatial

resolution was achieved, e.g. by synchrotron XCT. It has been recently stated that strain

gradients could affect the rate of growth of voids with radii smaller than the characteristic

length of the material [250] due to non-local plasticity effects. Approximate values of

characteristic lengths have been suggested to be in the range of 0.25-1 µm [1, 250], and

therefore void growth should be noticeably slowed down for void radii of 2 µm and smaller.

The smallest void volume considered in this study (∼ 200 µm3) corresponds to a radius of

3.6 µmwhich is still above the value where non-local plasticity effects are expected to occur.

This is also consistent with the fact that the majority of the pores with ∆V/V0 < 15%

(Figure 6.9(b)) have volumes well above 200 µm3, thus confirming that only internal gas

pressure can explain their small ∆V/V0.

6.3.2 Effect of the anisotropy on the variation of the volume and the

complexity factor of the pores

In the following, the effect of the anisotropy of the HCP lattice of Mg alloys on the

volume and CF changes is investigated. To this end, the isotropic model described in this

chapter (standard J2 plasticity with linear isotropic hardening model) and the CPFE

model presented in Chapter 4 are used to simulate the behavior of a polycrystalline

aggregate containing 1 and 2 pores.

Because the CPFE model was calibrated for a Mg AZ31 alloy with a fiber texture,

the same material parameters were chosen in the following simulations, but with an

6.3. Discussion 91

initial random texture. The compression stress-strain curve simulated for a fully-dense

polycrystal with 2,592 Voronoı grains (Figure 5.1) is illustrated in Figure 6.16. The

isotropic model was calibrated against this curve using the same mesh. Figure 6.16 shows

the corresponding calibrated stress-strain curve, in which, σ0 equal to 100 MPa and H

equal to 1.4 GPa. Young’s modulus and Poisson’s ratio values of 44.5 GPa and 0.35,

respectively, were used as in the previous model.

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.160

100

200

300

400

True strain

Tru

e s

tre

ss

(M

Pa

)

CPFE model

Isotropic model

Figure 6.16: Uniaxial compression true stress-true strain curves: CPFE model andstandard J2 plasticity with linear isotropic hardening model.

(a) (b)

Figure 6.17: FE mesh including: (a) 1 internal pore (equivalent to 3 Voronoı grains, Pore1) and (b) 2 internal pores (one equivalent to 3 Voronoı grains, Pore 1, and the secondequivalent to 1 Voronoı grain, Pore 2).

FE simulations were performed using ABAQUS/Explicit [251]. The behavior of the

pores was modeled as that of an ideal gas (air) at room temperature and ambient

pressure (density ρ=1.2041 kg/m3 and gas constant R=287.058 J/kgK). Both models

were subjected to a quasi-static hydrostatic pressure of 600 MPa, followed by relaxation.

Next, the FE mesh was modified to include: a) 1 internal pore equivalent to 3 Voronoı

grains (Pore 1), and b) 2 internal pores, one equivalent to 3 Voronoı grains (Pore 1)

92

and the second equivalent to 1 Voronoı grain (Pore 2). The FE meshes are shown in

Figures 6.17(a-b).

Tables 6.1 and 6.2 show the predicted changes in ∆V/V0 and ∆CF/CF0 for 1 and 2

pores, respectively. Table 6.1 shows that the predicted ∆V/V0 for 1 pore is similar for

both models. However, the ∆CF/CF0 values are slightly different, which indicates this

ratio seems to be affected by the anisotropy of plastic deformation in the neighboring

crystallographic orientations. This effect is, however, not very pronounced, due to the

rather large number of neighbors surrounding such a large pore.

Pore 1∆V/V0 ∆CF/CF0

Isotropic model 17.76% -17.15%

CPFE model 18.04% -19.65%

Table 6.1: ∆V/V0 and ∆CF/CF0 for the isotropic and the CPFE models with 1 internalpore.

Table 6.2 shows that the predictions corresponding to Pore 1 in the mesh with two

pores are very similar to those described above. However, Pore 2 is shown to be much

more affected by the anisotropy, as the differences between the ∆V/V0 and ∆CF/CF0

ratios predicted by both models are larger than for Pore 1.

Pore 1 Pore 2∆V/V0 ∆CF/CF0 ∆V/V0 ∆CF/CF0

Isotropic model 17.69% -17.63% 11.33% -9.46%

CPFE model 18.21% -19.98% 14.15% -13.28%

Table 6.2: ∆V/V0 and ∆CF/CF0 for the isotropic and CPFE models with 2 internalpores.

Note that these simulation results cannot quantitatively compare against the previous

results of this chapter (Figures 6.11 and 6.12) as the materials are different. However,

it can be concluded that a pore sees its volume change further increased and CF change

decreased by taking into account the local anisotropy of the polycrystal. This seems

to indicate that the population of pores not captured in the earlier simulations will be

captured by accounting for the crystal anisotropy of the polycrystal.

6.4 Conclusions

This study constitutes an experimental and modeling approach to investigate the evolution

of the porosity distribution and the mechanical behavior of an HPDC Mg AZ91 alloy

subjected to a hydrostatic pressure treatment. A total of 9,689 pores where tracked before

and after pressurization by image analysis based on XCT measurements. Conjointly, a

sub-volume of the material containing 200 cavities was simulated by FE analysis, using

6.4. Conclusions 93

the experimentally determined distribution of pores and an isotropic hardening for the

pore-free matrix. The main conclusions drawn from this investigation are listed below.

1. Pressurization results in a decrease of the volume of the pores. The distribution of

the relative volume change (∆V/V0) is bimodal and exhibits a first peak at small values of

approximately 6%, attributed to pores with an internal pressure arising from gas trapped

during processing, and a second one at 37%. ∆V/V0 values as high as 65% (for pores after

treatment with a volume of 200 µm3) are observed to take place in a large portion of some

belong to the first peak the pores with the smallest initial volumes considered (571 µm3).

2. The application of pressure also influences the pore geometry. Pores that belong

to the first peak of relative volume change (∆V/V0 ∼ 6%) tend to have positive values of

∆CF/CF0, i.e. they tend to become more regular after pressurization. ∆CF/CF0 tends

to subsequently decrease from positive values to negative ones (more irregular pores after

the pressure treatment) as ∆V/V0 increases. For pores for which ∆V/V0 is close to 27%

the average ∆CF/CF0 is close to 0%. As ∆V/V0 becomes larger than 40% the average

∆CF/CF0 adopts increasingly more negative values. To sum up, pores with small relative

volume changes tend to become more regular, and pores with large relative volume changes

tend to become more irregular.

3. The yield stress of the HPDC cast samples increases by about 31.5 MPa on

hydrostatic pressure treatment. Work hardening rates are, however, not seen to be affected

significantly.

4. FE simulations successfully capture the bimodal distribution of the relative volume

change after pressurization, albeit the average value of the second ∆V/V0 peak is lower

than the one measured experimentally. A good agreement between the distribution of

∆CF/CF0 is found between the XCT measurements and the simulations. Only the pores

undergoing pore size reductions higher than 45% (with final pore volumes smaller than

3,500 µm3), for which the ∆CF/CF0 adopts the largest negative values, are not captured

by the model. It is proposed that the source of this discrepancy resides in the inherent

plastic anisotropy of HCP Mg, that manifests itself especially when the pore size is of the

same order of magnitude than the grain size. The simple J2 plasticity model used here

cannot capture this phenomenon and a CPFE model with adequate grain orientation

is needed. Despite this drawback, the model successfully predicts quantitatively the

beneficial effect of the high pressure treatment on the yield stress of the HPDC casting.

5. Neighboring pores are observed to have a very strong effect on the evolution of

the porosity during the pressure treatment, so that the pores having close neighbors of

relatively larger size suffer a larger size reduction than isolated pores and/or pores whose

closest neighbors are relatively smaller. This is particularly true for the pore population

with the largest volumes, while they remained relatively insensitive for the smallest pores,

more influenced by the local crystal anisotropy.

CHAPTER 7

Conclusions and future work

7.1 Main conclusions

A CPFE model aimed at fully describing slip and twinning as well as their interactions

through hardening mechanisms was developed and implemented for Mg AZ31 alloy. The

proposed model was calibrated and validated against a set of quasi-static compression

tests at room temperature along RD and ND. The model was exploited by investigating

stress and strain fields, texture evolution, and slip and twin activities during deformation.

The flexibility of the overall model was ultimately demonstrated by casting light on an

experimental controversy on the role of the pyramidal 〈c+ a〉 slip vs. compression twinning

in the late stages of polycrystalline deformation, and a failure criterion related to basal

slip activity was proposed.

The developed CPFE model was expanded to represent a more realistic polycrystal

by use of 3D Voronoı tessellations with a large number of elements, allowing for the

high resolution study of the local intragranular mechanical fields. The robustness of the

extended model was validated against experimental data and compared to the previous

model. The new model exhibited quasi-identical averaged mesoscopic behavior as the

initial one. This implies that the previous simplified model is actually sufficient for

macro-/mesostructural properties (stress-strain curves, averaged twin/slip activity, etc.).

However, the new model was shown to be required for intragranular field studies. To this

end, a combined 3D EBSD and modeling approach was adopted to investigate the effect

of GB misorientation on tensile twin propagation in a rolled Mg AZ31 alloy sheet. For the

first time, 3D aspects of twin transfer across GBs were investigated. Both experiments and

simulations confirm that twin propagation is largely influenced by the GB misorientation

angle θ. GBs with low θ facilitate twin transfer. However, when θ increases, high local

stresses develop at the GBs to the extent that transferred twins are variants which do not

95

96

necessarily have the highest SFs with respect to the externally applied stress. Moreover,

twin morphology was found to depend on the SF: highest SF variants exhibit the well-

established plate morphology while low SF variants adopt irregular shapes.

A systematic characterization of the effect of pressure on the volume and the geometry

of individual pores was finally carried out for a HPDC Mg AZ91 alloy. Close to 10,000

pores were tracked before and after pressurization by XCT. The distribution of the relative

volume and CF changes was related to the initial pore volumes and to the presence or

absence of gas trapped as a consequence of the casting process. Additionally, it is shown

that a FE model using J2 plasticity with linear isotropic hardening for the pore-free matrix

successfully captures the observed changes in the distribution of the pore volume and CF

changes for a large majority of the pores. The small discrepancies between simulations and

experiments were shown to be related to the smallest pores, whose sizes are of the order of

the grain size and which are found to undergo the largest volume and CF changes. These

discrepancies were rationalized taking into account the anisotropy of the HCP lattice and

the influence of neighboring pores.

7.2 Future work

The following tasks are proposed as future work:

� Extension of the proposed CPFE model such that it takes into account both the

crystallographic texture and the real porosity distribution, which will allow for a

better prediction of the global mechanical behavior of Mg alloys.

� Incorporation of strain rate dependency and temperature effects into the developed

CPFE model.

� Incorporation of microstructural parameters, such as grain size and second phase

particle distributions into the developed CPFE model.

� Incorporation of the hardening effect of GNDs into the CPFE model.

� Extension of the proposed CPFE model to other HCP materials such as titanium or

zirconium, along with their respective phases.

� Extension of the combined 3D EBSD and modeling approach to investigate the

influence of alloying elements and testing conditions (strain rate, temperature) on

twin propagation in HCP metals.

APPENDIX A

Fractography analysis of a Mg AZ31 alloy

The fractured Mg AZ31 alloy sample compressed along RD at room temperature and at

10−4 s−1 is shown in Figure A.1 [252]. The angle between the compression axis and the

normal to the crack plane is close to 45°. Figure A.2 illustrates the SEM micrographs

of the fracture surfaces. It can be seen that the fracture mode is semibrittle or brittle

fracture and is characterized by transgranular cleavage and small scale microvoids.

Figure A.1: Fractured Mg AZ31 alloy specimen after uniaxial compression along RD [252].

97

98

(a)

(b)

Figure A.2: SEM micrographs illustrating the fracture surface of the Mg AZ31 alloy afteruniaxial compression along RD: (a) x500; (b) x1000.

APPENDIX B

Minimum deviation angle method

The {1012}〈1011〉 twinning system consists of six equivalent variants Tj |j=1...6. These are

summarized in Table B.1:

Tj (Plane) and [Direction]

T1 (1012) [1011]

T2 (1012) [1011]

T3 (1102) [1101]

T4 (1102) [1101]

T5 (0112) [0111]

T6 (0112) [0111]

Table B.1: Six {1012}〈1011〉 twin variants.

The determination of the active twin variant observed in a deformed grain Pi is carried

out by the method of the minimum θdev [92]. This method consists of the following three

steps.

First, the orientation matrix of each original grain, gPi, is calculated from the

corresponding Euler angles (ϕ1,Φ, ϕ2):

gPi=

(

cosϕ1 cosϕ2 − sinϕ1 sinϕ2 cos Φ sinϕ1 cosϕ2 + cosϕ1 sinϕ2 cos Φ sinϕ2 sinΦ

− cosϕ1 sinϕ2 − sinϕ1 cosϕ2 cosΦ − sinϕ1 sinϕ2 + cosϕ1 cosϕ2 cosΦ cosϕ2 sinΦ

sinϕ1 sinΦ − cosϕ1 sinΦ cos Φ

)

(B.1)

Secondly, the six possible twin rotations are applied to each gPito obtain the

orientation matrices corresponding to the six variants, gTij(Tij corresponds to variant

j of grain i).

gTij= Rj · gPi

(B.2)

99

100

Rj =

(

r21j(1 − cos θ) + cos θ r1jr2j(1− cos θ)− r3j sin θ r1jr3j(1− cos θ) + r2j sin θ

r2jr1j(1 − cos θ) + r3j sin θ r22j(1− cos θ) + cos θ r2jr3j(1− cos θ)− r1j sin θ

r3jr1j(1 − cos θ)− r2j sin θ r3jr2j(1− cos θ) + r1j sin θ r23j(1− cos θ) + cos θ

)

(B.3)

where r1j , r2j and r3j are the components of the twin direction corresponding to variant j

(rTj[xyz]) expressed in an orthonormal system and where θ=180° [64]. The calculation of

rTj[xyz]is carried out in the following way. The twin direction is commonly written using

the four Bravais-Miller index notation (rTj[uvtw]= ua1+va2+ta3+wc). A transformation

to a rhombohedral system (rTj[UVW]= Ua1 + V a2 +Wc) must first be performed, using

the relations:

U = u− t

V = v − t

W = w

(B.4)

Next, the transformation from the rhombohedral to the orthonormal coordinate system

in which subsequent calculations are carried out by Equation (B.4) is done by means of a

transformation matrix A:

A =

1 −12 0

0√32 0

0 0 ca

(B.5)

where c and a are the lattice parameters of the hexagonal lattice. Note that the matrix

A changes depending on how the hexagonal system is defined. In this case, the x-axis is

parallel to the a1-axis of a HCP crystalline structure. Therefore, the twin direction rTj[xyz]

in orthonormal coordinate system is given by:

rTj[xyz]= A · rTj[UVW]

(B.6)

Finally, the θdev values between the six calculated gTijand the orientation matrix of

the twins observed experimentally gTki|k=1...kobserved are estimated. The misorientation

matrix Mkj for each twin variant is calculated by:

Mkj = g−1Tik

· gTij(B.7)

θdevkj is obtained from:

cos θdevkj =(m11kj +m22kj +m33kj )− 1

2(B.8)

where m11kj , m22kj and m33kj correspond to the diagonal principal components of the

Mkj. The minimum θdevkj from all calculated twin variants, corresponds to the active

twin variant.

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List of Figures

1.1 Multiscale modeling framework of materials across the different length scales. 2

1.2 The HCP structure of Mg. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Main deformation mechanisms in Mg crystals. . . . . . . . . . . . . . . . . . 4

1.4 CRSS variations for the different slip/twin systems as a function of

temperature and strain rate for a Mg AZ31 alloy [50]. . . . . . . . . . . . . 5

1.5 Representation of the direct pole figures of a rolled Mg sheet. . . . . . . . . 6

1.6 Euler angles. The specimen coordinate XYZ is shown in black, the

crystallite system X’Y’Z’ is shown in red [64]. . . . . . . . . . . . . . . . . . 7

1.7 (a) Probability distribution vs. θ corresponding to an aggregate of HCP

crystals with basal and prismatic fiber textures. (b) Probability distribution

vs. θ of a random aggregate and of a basal fiber with a spread lower than

30° (adapted from [67]). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3.1 (a) X-ray diffractometer used in this work. (b) Schematic illustrating the

texture goniometer and the corresponding rotation angles. . . . . . . . . . . 20

3.2 Schematic drawing of the EBSD system. . . . . . . . . . . . . . . . . . . . . 21

3.3 (a) Origin of Kikuchi lines [64]. (b) Kikuchi pattern. . . . . . . . . . . . . . 21

3.4 Schematic representation of the grazing-incidence edge-milling method [192]. 22

3.5 (a) (FIB)-FEG-SEM dual beam instrument; (b) Schematic representation

of the tilt geometry [192]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.6 Schematic of a XCT acquisition configuration for cone beam geometry [205]. 25

3.7 Schematic of the X-ray absorption model used for tomographic reconstruc-

tion [201]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4.1 Multiplicative decomposition of the deformation gradient F. . . . . . . . . . 29

4.2 Uniaxial compression numerical tests along ND and RD. The resulting von

Mises stress field is plotted for both cases. . . . . . . . . . . . . . . . . . . . 33

123

124

4.3 Comparison between the experimental (top) and the numerical (bottom)

initial texture for the two cases: (a) along ND; (b) along RD. . . . . . . . . 34

4.4 Experimental and simulated stress-strain curves (true) corresponding to

uniaxial compression of the Mg AZ31 alloy: (a) along ND; (b) along RD. . 35

4.5 Comparison between {0001} experimental and numerical textures at defor-

mation state A, B, C, D, E and F for the compression along RD. . . . . . . 38

4.6 Normalized slip/twin activity for both cases: (a) compression along ND;

(b) compression along RD. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.7 Experimental and numerical twinning rotation fraction evolution. . . . . . . 40

4.8 Twinning rotation state in a longitudinal cut of the sample (blue: non-

rotated, red: rotated) for the states of deformation A, B, C, D, E and

F. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.9 Twinning rotation fraction as a function of the vertical element layer in the

sample at 4% deformation. . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.10 Corrected cumulated average slip/twin activity for both cases of compres-

sion: along RD and along ND. . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.11 Experimental and numerical stress-strain curves for the plane strain

compression along RD, constrained along TD. . . . . . . . . . . . . . . . . . 45

4.12 Comparison between {0001} experimental and numerical textures at defor-

mation state A, B, C, D and E for the plane strain compression along RD,

constrained along TD. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.13 Normalized slip/twin activity for the plane strain compression along RD,

constrained along TD. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.14 Corrected cumulated average slip/twin activity for the plane strain com-

pression along RD, constrained along TD; the dash line corresponds to the

acceptable range of values for the strain-to-failure. . . . . . . . . . . . . . . 47

5.1 Polycrystalline aggregate mesh of the proposed extended model. . . . . . . 51

5.2 (a) Comparison of experimental and simulated true stress-true strain curves

corresponding to uniaxial compression along RD and ND; (b) normalized

slip/twin activities for both numerical models (dashed line corresponds to

initial model described in Chapter 4). . . . . . . . . . . . . . . . . . . . . . 52

5.3 (a) Comparison of the cumulative twin volume fraction measured experi-

mentally and predicted by the initial model described in Chapter 4 and by

the extended model; (b) twin volume fraction predicted by the extended

model along three planes perpendicular to RD, ND and TD in the center

of the polycrystalline aggregate. . . . . . . . . . . . . . . . . . . . . . . . . . 53

List of Figures 125

5.4 Qualitative estimation of the twin volume fraction along three orthogonal

planes in the Mg AZ31 alloy tested in uniaxial compression along RD: (a)

volume reconstructed from three 2D EBSD maps measured in the sample

deformed up to 1.6%; (b) volume reconstructed from three 2D EBSD maps

measured in the sample deformed up to 2.6%; (c) twinned regions (dark)

predicted by the extended model in a sample deformed up to 5%. . . . . . . 54

5.5 Sample compressed along RD up to 2.6%. 3D reconstruction of inverse pole

figure maps on the RD obtained by the 3D EBSD method. P1, P2, P3, P4

refer to the four original grains, and Tij to variant j of grain i. . . . . . . . . 55

5.6 3D morphology of different twin variants: (a) T11 (volume size: 5 µm in TD

(blue axis)×10 µm in RD (green axis)×15.8 µm in ND (red axis)); (b) T12

(orange) and T13 (yellow) (volume size: 5 µm in TD×16.2 µm in RD×11 µm

in ND); (c) T41 (volume size: 5 µm in TD×8.8 µm in RD×8.2 µm in ND). . 57

5.7 Maximum twin thickness measured at different heights in grain P1: (a) for

each T11 lamella (the four lamella are named T111, T112, T113 and T114); (b)

for secondary (T12) and tertiary (T13) twins. . . . . . . . . . . . . . . . . . . 58

5.8 Variation in the crossing frequency with respect to the GB misorientation

angle measured in the sample compressed along RD up to a strain of 1.6%;

the arrows point to the θ between P1 and P2, P1 and P3, and, P1 andP4. . . 60

5.9 2D EBSD maps in the RD of the GB between grains P1 and P2; map (a)

corresponds to a height of 3.4 µm and map (b) to a height of 4.4 µm. . . . . 61

5.10 Sequence of 2D EBSD maps of the GB between grains P1 and P3 in height

steps of 200 nm; the first slice shown (a) corresponds to a height of 1.6 µm

and the final slice (o) to a height of 4.4 µm. . . . . . . . . . . . . . . . . . . 62

5.11 Sequence of 2D EBSD maps in the RD at height steps of 200 nm, which

illustrate the GB between grains P1 and P4; the first slice shown (a)

corresponds to a height of 1 µm and the final slice (h) to a height of 2.4 µm. 63

5.12 2D EBSD section of the volume illustrated in Figure 5.5, located at a height

of 4.8 µm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

5.13 (a) Polycrystalline model (a quarter of the model is hidden to ease the

visualization of the two grains of interest) consisting of a 25-grains Voronoı

tessellation (2,113,409 tetrahedral elements); the two grains of interest are

assigned the orientations of the corresponding P1 − Pi |i=2,3,4 pairs and

cut transversally for visualization purposes of: (b) twin transfer across the

P1 − P2 boundary (3.9% of strain); (c) twin transfer across the P1 − P3

boundary (6% of strain); (d) twin transfer across the P1−P4 boundary (6%

of strain); the red line shows the paths from grain P1 to the neighboring

grains along which slip activity was evaluated in Figure 5.14. . . . . . . . . 66

5.14 Normalized slip/twin activities in the vicinity of the three GBs P1 − P2,

P1 −P3 and P1 −P4 at 2.8% and 5%, respectively, along the path drawn in

red in Figures 5.13(b-d); the black dashed line indicates the GB. . . . . . . 67

126

5.15 (a) Polycrystalline model cut transversally for visualization purposes

showing the twin nucleation point at the low angle P1 − P2 GB and (b)

variation of the von Mises stress along the path drawn in red in (a) at

a strain of 2.1%; (c) polycrystalline model and (d) normalized slip/twin

activity along the path drawn in red in (c) at a strain of 2.5%). . . . . . . 69

5.16 Polycrystalline model cut transversally for visualization purposes at a strain

of 0.7% showing grains P1 and P4 with the fields of (a) the von Mises stress

and (b) the activity of the primary variant in grain P1 (the black arrow

indicates the point of nucleation). . . . . . . . . . . . . . . . . . . . . . . . 69

6.1 Sample geometry of the Mg AZ91 alloy dog-bone shaped casting fabricated

by HPDC. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

6.2 Microstructure of the as-HPDC Mg AZ91 alloy at different magnifications;

the micrographs were taken in Grip 1. . . . . . . . . . . . . . . . . . . . . . 72

6.3 (a) Schematic representation of CF parameter: a cavity (in blue) is

approximated by its equivalent ellipsoid (in red) and a, b and c are the

axes of the equivalent ellipsoid; (b) CF parameter definition [201]. . . . . . 74

6.4 FE mesh of the selected tomographic sub-volume geometry: a sector of the

pore-free mesh was removed for visualization purposes. . . . . . . . . . . . . 76

6.5 Uniaxial compression true stress-true strain curves: experimental data and

model calibration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

6.6 Low resolution tomographic reconstruction of the complete dog-bone

casting, high resolution reconstruction (initial state) of the cylindrical

volume subjected to the pressure treatment and of the sub-volume of

300×300×300 µm3 used in the FE analysis. . . . . . . . . . . . . . . . . . . 78

6.7 Reconstructed cross-sectional views of the same slice before (a) and after (b)

the pressure treatment; magnified region at the largest area of the central

pore before (c) and after (d) the pressure treatment. . . . . . . . . . . . . . 78

6.8 3D reconstruction of the sub-volume extracted for simulation purposes

before (a) and after (b) the pressure treatment; (c) superposition of volumes

from (a) and (b) with initial pores (before treatment) set to semitransparent. 79

6.9 (a) Histogram of the relative volume change (∆V/V0) of the 9,689 tracked

cavities. (b) Pore volume after pressure treatment vs. relative volume change. 81

6.10 (a) Histogram of the relative CF change (∆CF/CF0) of the 9,689 tracked

cavities; (b) relative CF change (∆CF/CF0) vs. relative volume change

(∆V/V0) (positive values of ∆CF/CF0 indicate transition to more regular

pores, while negative values indicate transition to more irregular pores). . . 82

6.11 Comparison between experimental and simulations results with and without

internal pressure for the same sub-volume; (a) normalized number of pores

vs. the relative volume change (∆V/V0); (b) pore volume after pressure

treatment vs. relative volume change (∆V/V0). . . . . . . . . . . . . . . . . 84

List of Figures 127

6.12 Comparison of experimental results and simulations with and without

internal pressure for the same sub-volume; (a) normalized number of

pores vs. the relative CF change (∆CF/CF0); (b) relative CF change

(∆CF/CF0) vs. relative volume change (∆V/V0) . . . . . . . . . . . . . . . 85

6.13 Uniaxial compression response of (a) cylinders extracted from Grip 1 before

and after the pressure treatment as measured experimentally and (b) the

simulated sub-volume. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

6.14 Examples of the equivalent plastic strain field located around the pores after

releasing the hydrostatic pressure. . . . . . . . . . . . . . . . . . . . . . . . 89

6.15 Relative volume pore change (∆V/V0) vs. Rw = w/(w + rNN ) for several

initial pore volume classes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

6.16 Uniaxial compression true stress-true strain curves: CPFE model and

standard J2 plasticity with linear isotropic hardening model. . . . . . . . . 91

6.17 FE mesh including: (a) 1 internal pore (equivalent to 3 Voronoı grains, Pore

1) and (b) 2 internal pores (one equivalent to 3 Voronoı grains, Pore 1, and

the second equivalent to 1 Voronoı grain, Pore 2). . . . . . . . . . . . . . . . 91

A.1 Fractured Mg AZ31 alloy specimen after uniaxial compression along RD [252]. 97

A.2 SEM micrographs illustrating the fracture surface of the Mg AZ31 alloy

after uniaxial compression along RD: (a) x500; (b) x1000. . . . . . . . . . . 98

List of Tables

1.1 Planes and directions of main deformation mechanisms in Mg and Mg alloys. 3

4.1 Composition of the Mg AZ31 alloy under study (in wt. %). . . . . . . . . . 27

4.2 Strain at transition states. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.3 Material parameters after calibration. . . . . . . . . . . . . . . . . . . . . . 36

4.4 Material parameters sensibility. . . . . . . . . . . . . . . . . . . . . . . . . . 37

4.5 Transition strain between regions for the plane strain compression case. . . 45

5.1 SF and θdev corresponding to all the twins contained in the volume

of Figure 5.5; the values of the Euler angles (ϕ1,Φ, ϕ2 – in degrees)

corresponding to the original grains (Pi) and the twins (Tij) are included.

The minimum θdev values are highlighted in bold letters. . . . . . . . . . . . 56

6.1 ∆V/V0 and ∆CF/CF0 for the isotropic and the CPFE models with 1

internal pore. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

6.2 ∆V/V0 and ∆CF/CF0 for the isotropic and CPFE models with 2 internal

pores. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

B.1 Six {1012}〈1011〉 twin variants. . . . . . . . . . . . . . . . . . . . . . . . . . 99

129

List of Publications

Journal Publications

� F. Sket, A. Fernandez, A. Jerusalem, J. Molina-Aldareguia and M.T. Perez-

Prado. Effect of hydrostatic pressure on the 3D porosity distribution and mechanical

behavior of a high pressure die-cast Mg AZ91 alloy. Journal of the Mechanics

and Physics of Solids, Under review.

� A. Fernandez, A. Jerusalem, I. Gutierrez-Urrutia, M.T. Perez-Prado. Three-

dimensional investigation of the grain boundary-twin interactions in a Mg AZ31 alloy

by electron backscatter diffraction and continuum modeling. Acta Materialia,

61:7679-7692, 2013.

� A. Jerusalem, A. Fernandez, A. Kunz and J.R. Greer. Continuum modeling

of dislocation starvation and subsequent nucleation in nano-pillar compression.

Scripta Materialia, 66 (2):93-96, 2012.

� A. Fernandez, M.T. Perez-Prado, Y. Wei, A. Jerusalem. Continuum modeling

of the response of a Mg alloy AZ31 rolled sheet during uniaxial deformation.

International Journal of Plasticity, 27:1739-1757, 2011.

� A. Jerusalem, A. Fernandez, M.T. Perez-Prado. Continuum modeling of {1012}

twinning in a Mg-3wt.%Al-1wt.%Zn rolled sheet. Revista de Metalurgia de

Madrid, Invited paper, 7:133-137, 2010.

Conferences

� Magnesium Workshop Madrid 2013, Madrid, Spain (2013). A. Fernandez,

I. Gutierrez-Urrutia, A. Jerusalem and M.T. Perez-Prado. 3D polycrystalline

continuum model of deformation mechanisms in a rolled magnesium AZ31 alloy.

(Oral presentation)

131

132

� TMS 2013 142nd Annual Meeting Exhibition, San Antonio, TX, USA (2013). A.

Fernandez, F. Sket, J. Molina-Aldareguia, M.T. Perez-Prado and A. Jerusalem.

Influence of hydrostatic pressure on porosity of die-cast Mg alloys: experimental

and numerical studies. (Poster)

� 8th European Solid Mechanics Conference ESMC2012, Graz, Austria (2012). A.

Fernandez, F. Sket, J. Molina-Aldareguia, M.T. Perez-Prado and A. Jerusalem.

Influence of hydrostatic pressure on porosity of die-cast Mg alloys: experimental

and numerical studies. (Oral presentation)

� 8th European Solid Mechanics Conference ESMC2012, Graz, Austria (2012). A.

Jerusalem, A. Fernandez, A. Kunz and J.R. Greer. Continuum modeling of

dislocation starvation and subsequent nucleation in nano-pillar compressions. (Oral

presentation)

� International Conference on Computational Plasticity COMPLAS XI, Barcelona,

Spain (2011). A. Fernandez, M.T. Perez-Prado, Y. Wei and A. Jerusalem. Con-

tinuum modeling of Mg alloy AZ31 under uniaxial deformation. (Oral presentation)

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