Microstructure-based numerical modeling of the mechanical...

Universidad Politecnica de Madrid

Escuela Tecnica Superior deIngenieros de Caminos, Canales y Puertos

Microstructure-based numericalmodeling of the mechanical behavior

of Mg alloys

Tesis doctoral

Vicente Herrera SolazIngeniero de Caminos

2015

Departamento de Ciencia de Materiales

Escuela Tecnica Superior de Ingenieros deCaminos, Canales y Puertos

Universidad Politecnica de Madrid

Microstructure-based numericalmodeling of the mechanical behavior

of Mg alloys

Tesis doctoral

Vicente Herrera SolazIngeniero de Caminos

Directores de la tesis

Javier Segurado EscuderoDr. Ingeniero de Materiales

Profesor Titular de Universidad

Javier Llorca MartınezDr. Ingeniero de Caminos, Canales y Puertos

Catedratico de Universidad

2015

Tribunal nombrado por el Sr. Rector Magfco. de la Universidad Politécnica de Madrid, el día...............de.............................de 20....

Presidente:

Vocal:

Vocal:

Vocal:

Secretario:

Suplente:

Suplente: Realizado el acto de defensa y lectura de la Tesis el día..........de........................de 20 ... en la E.T.S.I. /Facultad.................................................... Calificación .................................................. EL PRESIDENTE LOS VOCALES

EL SECRETARIO

Contents

Agradecimientos III

Resumen V

Acknowledgments VII

Abstract IX

Notation XI

1 Introduction 1

1.1 Importance of Mg alloys . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Deformation mechanisms of Mg alloys . . . . . . . . . . . . . . . . . . . . . 5

1.3 Modeling of polycrystal behavior . . . . . . . . . . . . . . . . . . . . . . . 11

1.3.1 Representative volume element . . . . . . . . . . . . . . . . . . . . 12

1.3.2 Crystal plasticity model . . . . . . . . . . . . . . . . . . . . . . . . 13

1.3.3 Mean-field approximations . . . . . . . . . . . . . . . . . . . . . . . 16

1.3.4 Computational homogenization . . . . . . . . . . . . . . . . . . . . 20

1.4 Mechanical behavior of single crystals . . . . . . . . . . . . . . . . . . . . . 21

1.5 Objectives and structure of the thesis . . . . . . . . . . . . . . . . . . . . . 24

2 Models and algorithms 27

2.1 Finite element crystal plasticity model . . . . . . . . . . . . . . . . . . . . 27

2.2 Crystal plasticity model for Mg alloys . . . . . . . . . . . . . . . . . . . . . 28

2.2.1 Time discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.2.2 Subroutine parameters and outputs . . . . . . . . . . . . . . . . . . 38

2.3 Computational homogenization framework . . . . . . . . . . . . . . . . . . 41

I

CONTENTS

2.3.1 Microstructure representation . . . . . . . . . . . . . . . . . . . . . 42

2.4 Inverse optimization strategy . . . . . . . . . . . . . . . . . . . . . . . . . 45

3 Results and discussion 51

3.1 AZ31 Mg alloy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.1.1 Material and processing . . . . . . . . . . . . . . . . . . . . . . . . 51

3.1.2 Mechanical behavior . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.1.3 Optimization strategy and results . . . . . . . . . . . . . . . . . . . 53

3.1.4 Influence of the input information . . . . . . . . . . . . . . . . . . . 63

3.1.5 Influence of the initial set of parameters . . . . . . . . . . . . . . . 67

3.2 Mg alloys containing rare earths . . . . . . . . . . . . . . . . . . . . . . . . 68

3.2.1 Materials and processing . . . . . . . . . . . . . . . . . . . . . . . . 70



3.3 MN11 Mg alloy at different temperatures . . . . . . . . . . . . . . . . . . . 78

3.3.1 Material and processing . . . . . . . . . . . . . . . . . . . . . . . . 79



4 Conclusions and future work 95

4.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

4.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

A Crystal properties 99

Bibliography 101

List of Figures 115

List of Tables 121

B Personal contributions 123

II

Agradecimientos

En primer lugar agradecer a mis co-tutores D. Javier Segurado y D. Javier LLorca el

que depositaran su confianza en mı para la realizacion de la presente Tesis. Para mı ha

sido un privilegio trabajar bajo su supervision, tanto por el aspecto humano como por el

academico. El desarrollo de la tesis ha sido un camino perfectamente guiado en los aspectos

teoricos por D. Javier Segurado ademas de rigurosamente planificado y estructurado por

D. Javier LLorca. Sin el toque magistral de ambos, sin duda, no hubiese podido alcanzar

el objetivo.

En segundo lugar mostrar mi gratitud a todo el personal del Departamento de Ciencia

de Materiales de la E.T.S. de Ingenieros de Caminos de la U.P.M, desde mis companeros

mas cercanos de la “zona comun”, Maricely, Monica, Daniel, Conchi, Chao y Mariangel,

hasta los profesores del departamento, tecnicos del laboratorio y personal de adminis-

tracion, por su trato amable y disposicion en todo momento.

No puedo soslayar la oportunidad de estancia que se me ofrecio en la Michigan State

University. Gracias a D. Carl Boelhert por brindarmela y por supuesto a Ajith Chakkedath

por ensenarme los entresijos del microscopio electronico de barrido y de la difraccion de

electrones retrodispersados.

Remarcar que la investigacion realizada en esta tesis doctoral se ha realizado en el

marco del proyecto de investigacion “Analisis de la evolucion microestructural y del com-

portamiento mecanico de aleaciones de Mg-Mn-RE” entre la Michigan State University, el

Instituto IMDEA Materiales y la Universidad Politecnica de Madrid, dentro de la Materials

World Network. La investigacion de los equipos espanoles ha sido financiada por el Minis-

terio de Economıa y Competitividad dentro del programa Nacional de Internacionalizacion

de la I+D (proyecto PRI-PIBUS-2011-0990). Sin este apoyo institucional, sin duda, no

habrıa sido posible todo el presente trabajo.

Por ultimo, reconocer y dar las gracias a mi mujer Angela, por apoyarme en la decision

III

Agradecimientos

de doctorarme, ası como por su comprension y aliento en los momentos mas difıciles.

IV

Resumen

Dentro de los materiales estructurales, el magnesio y sus aleaciones estan siendo el foco

de una de profunda investigacion. Esta investigacion esta dirigida a comprender la relacion

existente entre la microestructura de las aleaciones de Mg y su comportamiento mecanico.

El objetivo es optimizar las aleaciones actuales de magnesio a partir de su microestructura

y disenar nuevas aleaciones. Sin embargo, el efecto de los factores microestructurales (como

la forma, el tamano, la orientacion de los precipitados y la morfologıa de los granos) en el

comportamiento mecanico de estas aleaciones esta todavıa por descubrir.

Para conocer mejor de la relacion entre la microestructura y el comportamiento mecanico,

es necesaria la combinacion de tecnicas avanzadas de caracterizacion experimental como

de simulacion numerica, a diferentes longitudes de escala. En lo que respecta a las tecnicas

de simulacion numerica, la homogeneizacion policristalina es una herramienta muy util

para predecir la respuesta macroscopica a partir de la microestructura de un policristal

(caracterizada por el tamano, la forma y la distribucion de orientaciones de los granos) y el

comportamiento del monocristal. La descripcion de la microestructura se lleva a cabo me-

diante modernas tecnicas de caracterizacion (difraccion de rayos X, difraccion de electrones

retrodispersados, ası como con microscopia optica y electronica). Sin embargo, el compor-

tamiento del cristal sigue siendo difıcil de medir, especialmente en aleaciones de Mg, donde

es muy complicado conocer el valor de los parametros que controlan el comportamiento

mecanico de los diferentes modos de deslizamiento y maclado.

En la presente tesis se ha desarrollado una estrategia de homogeneizacion computacional

para predecir el comportamiento de aleaciones de magnesio. El comportamiento de los

policristales ha sido obtenido mediante la simulacion por elementos finitos de un volumen

representativo (RVE) de la microestructura, considerando la distribucion real de formas

y orientaciones de los granos. El comportamiento del cristal se ha simulado mediante

un modelo de plasticidad cristalina que tiene en cuenta los diferentes mecanismos fısicos

V

Resumen

de deformacion, como el deslizamiento y el maclado. Finalmente, la obtencion de los

parametros que controlan el comportamiento del cristal (tensiones crıticas resueltas (CRSS)

ası como las tasas de endurecimiento para todos los modos de maclado y deslizamiento) se

ha resuelto mediante la implementacion de una metodologıa de optimizacion inversa, una de

las principales aportaciones originales de este trabajo. La metodologıa inversa pretende,

por medio del algoritmo de optimizacion de Levenberg-Marquardt, obtener el conjunto

de parametros que definen el comportamiento del monocristal y que mejor ajustan a un

conjunto de ensayos macroscopicos independientes. Ademas de la implementacion de la

tecnica, se han estudiado tanto la objetividad del metodologıa como la unicidad de la

solucion en funcion de la informacion experimental.

La estrategia de optimizacion inversa se uso inicialmente para obtener el compor-

tamiento cristalino de la aleacion AZ31 de Mg, obtenida por laminado. Esta aleacion

tiene una marcada textura basal y una gran anisotropıa plastica. El comportamiento de

cada grano incluyo cuatro mecanismos de deformacion diferentes: deslizamiento en los

planos basal, prismatico, piramidal hc+ai, junto con el maclado en traccion. La validez de

los parametros resultantes se valido mediante la capacidad del modelo policristalino para

predecir ensayos macroscopicos independientes en diferentes direcciones.

En segundo lugar se estudio mediante la misma estrategia, la influencia del contenido de

Neodimio (Nd) en las propiedades de una aleacion de Mg-Mn-Nd, obtenida por extrusion.

Se encontro que la adicion de Nd produce una progresiva isotropizacion del comportamiento

macroscopico. El modelo mostro que este incremento de la isotropıa macroscopica era

debido tanto a la aleatoriedad de la textura inicial como al incremento de la isotropıa del

comportamiento del cristal, con valores similares de las CRSSs de los diferentes modos de

deformacion.

Finalmente, el modelo se empleo para analizar el efecto de la temperatura en el com-

portamiento del cristal de la aleacion de Mg-Mn-Nd. La introduccion en el modelo de los

efectos non-Schmid sobre el modo de deslizamiento piramidal hc+ai permitio capturar el

comportamiento mecanico a temperaturas superiores a 150◦C. Esta es la primera vez, de

acuerdo con el conocimiento del autor, que los efectos non-Schmid han sido observados en

una aleacion de Magnesio.

VI

Acknowledgments

I want to thank my advisors Dr. Javier Segurado and Dr. Javier LLorca for trusting

me to carry out this thesis. It has been a privilege to work under their supervision, in both

human and academic aspects. The development of the thesis has been a path perfectly

guided by Javier Segurado, on the theoretical issues, as well as rigorously planned and

perfectly polished by Javier Llorca. Without the masterstroke of both, I would not have

been able to achieve the goal.

I also want to express my appreciation to all the staff of the Department of Materials

Science of the Civil Engineering School of the Polytechnic University of Madrid, from

my closest colleagues in the common area: Maricely, Monica, Daniel, Conchi, Chao and

Mariangel, to professors of the department, laboratory technicians and administrative staff,

both for their kind treatment and for their availability at any time.

I cannot ignore my stage at Michigan State University. Thanks to Dr. Carl Boelhert

for his support and of course to Ajith Chakkedath for teaching me the intricacies of the

scanning electron microscope and of the electron backscatter diffraction analysis.

I have to acknowledge that the research in this thesis was carried out in the framework

of the research project “Analysis of the microstructural evolution and mechanical behavior

of Mg-Mn-rare earth alloys”, carried out by Michigan State University, IMDEA Materials

Institute and the Polytechnic University of Madrid within the Materials World Network.

The Spanish research has been funded by the Spanish Ministry of Economy and Competi-

tiveness within the National program of internationalization for research and development

(project PRI-PIBUS-2011-0990). This work certainly would not have been all possible

without this institutional support.

Finally, to acknowledge and thank my wife Angela, for supporting me in my decision

of getting a PhD, as well as, for her encouragement in the most difficult moments.

VII

Abstract

The study of Magnesium and its alloys is a hot research topic in structural materials.

In particular, special attention is being paid in understanding the relationship between mi-

crostructure and mechanical behavior in order to optimize the current alloy microstructures

and guide the design of new alloys. However, the particular effect of several microstructural

factors (precipitate shape, size and orientation, grain morphology distribution, etc.) in the

mechanical performance of a Mg alloy is still under study.

The combination of advanced characterization techniques and modeling at several

length scales is necessary to improve the understanding of the relation microstructure

and mechanical behavior. Respect to the simulation techniques, polycrystalline homog-

enization is a very useful tool to predict the macroscopic response from polycrystalline

microstructure (grain size, shape and orientation distributions) and crystal behavior. The

microstructure description is fully covered with modern characterization techniques (X-ray

diffraction, EBSD, optical and electronic microscopy). However, the mechanical behavior

of single crystals is not well-known, especially in Mg alloys where the correct parame-

terization of the mechanical behavior of the different slip/twin modes is a very difficult

task.

A computational homogenization framework for predicting the behavior of Magnesium

alloys has been developed in this thesis. The polycrystalline behavior was obtained by

means of the finite element simulation of a representative volume element (RVE) of the

microstructure including the actual grain shape and orientation distributions. The crystal

behavior for the grains was accounted for a crystal plasticity model which took into account

the physical deformation mechanisms, e.g. slip and twinning. Finally, the problem of the

parametrization of the crystal behavior (critical resolved shear stresses (CRSS) and strain

hardening rates of all the slip and twinning modes) was obtained by the development of an

inverse optimization methodology, one of the main original contributions of this thesis. The

IX

Abstract

inverse methodology aims at finding, by means of the Levenberg-Marquardt optimization

algorithm, the set of parameters defining crystal behavior that best fit a set of independent

macroscopic tests. The objectivity of the method and the uniqueness of solution as function

of the input information has been numerically studied.

The inverse optimization strategy was first used to obtain the crystal behavior of a rolled

polycrystalline AZ31 Mg alloy that showed a marked basal texture and a strong plastic

anisotropy. Four different deformation mechanisms: basal, prismatic and pyramidal hc+aislip, together with tensile twinning were included to characterize the single crystal behavior.

The validity of the resulting parameters was proved by the ability of the polycrystalline

model to predict independent macroscopic tests on different directions.

Secondly, the influence of Neodymium (Nd) content on an extruded polycrystalline

Mg-Mn-Nd alloy was studied using the same homogenization and optimization framework.

The effect of Nd addition was a progressive isotropization of the macroscopic behavior. The

model showed that this increase in the macroscopic isotropy was due to a randomization

of the initial texture and also to an increase of the crystal behavior isotropy (similar values

of the CRSSs of the different modes).

Finally, the model was used to analyze the effect of temperature on the crystal behavior

of a Mg-Mn-Nd alloy. The introduction in the model of non-Schmid effects on the pyramidal

hc+ai slip allowed to capture the inverse strength differential that appeared, between the

tension and compression, above 150◦C. This is the first time, to the author’s knowledge,

that non-Schmid effects have been reported for Mg alloys.

X

Notation

Throughout the thesis the tensor notation will be used as detailed below

a Vector, components ai

α Second order tensor, components αij

A Fourth order tensor, components Aijkl

I Indentity tensor

AT Transposed tensor

ab Scalar product , (ab) = aibi

a� b Vectorial product

a b (a b)ij = aibj

αa (αa)i = αijaj

Aα (Aα)ij = Aijklαkl

αβ (αβ)ij = αikβkj

α : β (αβ) = αijβij

AB (AB)ijkl = AijmnBmnkl

A : B (A : B) = AijklBijkl

α β (α β)ijkl = αijβkl

XI

Notation

The main variables used throughout the thesis are detailed in the following list

hai: a directions in HCP crystals (in the basal plane)

hci: c directions in HCP crystals (normal to the basal plane)

ha+ ci: a + c directions in HCP crystals

C: Fourth order elastic stiffness tensor

Cα: Fourth order elastic stiffness tensor reoriented after twinning

S: Second Piola-Kirchhoff stress tensor

σ: Cauchy stress tensor

Ee: Green elastic strain tensor

m∗: Schmid factor

a1, a2, a3, c: Axes that define the HCP crystallographic structure

CRSS, ταc : Critical resolved shear stress on the system α

τα0,c: Initial value of the critical resolved shear stress on the system α

ταsat: Saturation value of the critical resolved shear stress on the system α

h0: Initial tangent modulus

qi,j: Matrix describing the latent hardening of a crystal

τα: Resolved shear stress on the system α

n, s: Plane normal and slip direction corresponding to a certain slip plane

F: Deformation gradient tensor

Fe: Elastic part of the deformation gradient tensor

Fp: Plastic part of the deformation gradient tensor

L: Total velocity gradient tensor

Le: Elastic velocity gradient

Lp: Plastic velocity gradient

Lpsl: Plastic velocity gradient related due to slip

Lptw: Plastic velocity gradient related due to twinning

Lpre−sl: Plastic velocity gradient related due to re-slip

Nsl: Number of slip systems

Ntw: Number of twinning systems

Nsl−tw: Number of slip systems that can undergo re-slip

Nre−slip: Number of re-slip systems

XII

Notation

γi: Plastic shear rate on the slip system i

γ0: Reference shear strain rate

γtw: Characteristic shear of the twinning mode

m: Rate-sensitivity exponent

fα: Rate of the volume fraction transformation on the twin system α

f0: Reference twinning rate

Qα: Rotation tensor

fα: Volume fraction of twinned material on the twin system α

i,α: Integer numbers used to define the slip and twin sytems respectively

R(Fe): Tensorial residual function depending on the elastic deformation gradient

Fe

J: Fourth order tensor corresponding to Jacobian obtained as ∂R(Fe)∂Fe

ϕ1, φ and ϕ2: Euler angles defining the rotation of the global reference system to obtain

the crystal reference system

O(β): Objective error function depending on the set of β parameters

J: Jacobian matrix in Levenberg-Marquardt algorithm

λ: Dumping parameter in the linear set of equations in Levenberg-Marquardt

algorithm

β: Set of parameters to be obtained by means of Levenberg-Marquardt

algorithm

xi, yi: Set of n points defining some experimental result

xi, y∗i : Set of n points defining some model prediction corresponding to some

experimental result

η: Non Schmid tensor

XIII

Notation

The main Acronyms used are detailed in the following list

EBSD: Electron BackScatter Diffraction

SEM: Scanning Electron Microscope

RVE: Representative Volume Element

CRSS: Critical Resolved Shear Stress

HCP: Hexagonal Closed Packed

FCC: Face Centered Cubic

BCC: Body Centered Cubic

RD: Rolling Direction

ND: Normal Direction to Rolling Direction

TD: Transverse Direction normal to RD and ND

ED: Extrusion Direction

AZ31: Magnesium alloy containing 3% Al and 1% Zn in wt

RE: Rare earths

MN10: Magnesium alloy containing 1% Mn and 0.5% RE(Nd) in wt

MN11: Magnesium alloy containing 1% Mn and 1% RE(Nd) in wt

XIV

Chapter 1Introduction

1.1 Importance of Mg alloys

The increasing demand for economical use of limited energy resources and the control

over emissions to lower environmental impact have acted as driving forces to introduce

lighter materials in transport. Mg is, obviously, a promising option due to the combi-

nation of low density and good mechanical properties. Mg is the sixth most abundant

element in the earth’s crust, representing 2.7% of the earth’s crust [Okamoto, 1998]. Mg

compounds can be found worldwide and the most common compounds are magnesite

(MgCO3), dolomite (MgCO3� CaCO3), carnallite (KCl � MgCl2� 6H2O). Mg is also found

in seawater [Avedesian and Baker, 1999].

Mg is the lightest of all structural metals, with a density of 1.74 g/cm3, and the third

most-commonly used structural-metal, following steel and Al [Pekguleryuz et al., 2013].

Because of its low density, Mg alloys are excellent candidates for weight-critical applica-

tions. The elastic modulus of polycrystalline Mg is 45 GPa, leading to a specific stiffness

similar to that of Al and Ti, but it presents limited ductility, strength and creep resistance

and these limitations hinder its widespread use in structural applications [Alam et al.,

2011].

Mg is chemically active and can react with other metallic alloying elements to form

intermetallic compounds. These intermetallic phases are found in all Mg alloys, modifying

the microstructure, and hence, the mechanical properties. An extensive review of the most

common alloying elements in Mg can be found in [Avedesian and Baker, 1999, Lyon et al.,

1

Chapter 1. Introduction

2005, Gupta et al., 2011], and a summary of the most relevant elements is presented below.

� Al is one of the most common alloying elements. Addition of Al results in the

enhancement of hardness and strength. It also improves castability. The alloys with more

than 6 wt% of aluminum can be heat treated.

� The addition of Mn enhances the saltwater corrosion resistance of Mg-Al and Mg-

Al-Zn alloys. The low solubility of Mn in Mg limits the amount of Mn that can be added.

Mn is usually incorporated with other alloying elements like aluminum.

� Rare earths, as Nd, Ce, La, Yt, are added to increase the strength (specially at high

temperature), creep and corrosion resistance. Furthermore, it has been observed [Herrera-

Solaz et al., 2014a] that the RE additions have an effect on the recrystallization process

after forming, leading to more random textures. Their use is limited to high-added value

applications, as rare earths are expensive.

� Zn is usually used together with Al to increase the strength without reducing ductility.

Moreover, the presence of Zn with Ni and Fe impurities can also assist to improve the

corrosion resistance.

� Zr acts as an excellent grain refiner when incorporated into alloys containing Zn, Th,

rare earths, or a combination of these elements. However, it cannot be used with Al or Mn

because of the formation of stable intermetallic compounds with these alloying elements.

The mechanical properties of Mg alloys can be greatly improved by adding alloying

elements by means of solid solution and precipitation hardening. As a result, Mg alloys

are currently used in non-structural applications in different sectors including automotive,

aerospace, health care, sports, electronics, etc. [Gupta et al., 2011]. Regarding automotive

applications, Mg has been used in commercial vehicles since the 1930s in the Volkswagen

Beetle, that already contained more than 20 kg of Mg alloys in the transmission housing and

the crankcase. More recently, the environmental and legislative pressure on the automotive

industry to produce lighter and more efficient vehicles have resulted in the surge of the use

of light alloys. Widely used conventional steel parts are being replaced by new advanced

materials such as Mg, Al, and metal-matrix composites. The most common applications

of Mg alloys can be found in parts without structural responsibility like wheel assembly,

gearbox housing and steering wheels (Fig. 1.1(a)), and they are starting to be used in

interior parts such as the seat frame (Fig. 1.1(b)), steering column housing, driver’s airbag

housing, and lock body [Kim and Han, 2008].

With respect to aerospace applications, weight reduction is one of the most critical

2

1.1 Importance of Mg alloys

(a) (b)

(c) (d)

(e)

Figure 1.1: Steering wheel of the US Toyota Camry (a), Faurecia’s front seat frame

platforms developed and produced for Nissan, General Motors and Volkswagen (b), Boe-

ing 737 thrust reverser (c), Toshiba Portege Z830 � 104 with magnesium alloy chassis

(d), Bike with a frameset and wheels that are injection metal molded in Mg (e)

objectives due to the increasing need for emission reduction and fuel efficiency. Over the

years, Mg alloys have been used in both civil and military aircraft. Some applications

3


include the thrust reversal (for several Boeing, Fig. 1.1(c)), gearbox (Rolls-Royce), and

helicopter transmission casings. Mg alloys are becoming increasingly attractive for the

aerospace industry after the recent release by the Federal American Aviation (FAA) of a

report required to get Mg in the cabin of aircraft under special conditions approvals. How-

ever, its application in structural parts replacing Ti or Al alloys is still under investigation.

Mg alloys have also been used in health care. They were introduced as orthopedic

biomaterials in the first half of the last century [Earl D. Mcbride, 1938]. However, its

use has been limited because of its low corrosion resistance. Despite this, Mg is very

attractive for bone replacement in biomedical applications because its elastic modulus,

compressive strength and density are closer to those of natural bone than any other metallic

material [Staiger et al., 2006], while its fracture toughness is much higher than that of

hydroxyapatite. In addition, Mg has good biocompatibility and it is biodegradable in

human body fluid by corrosion, thus eliminating the need for another operation to remove

the implant. All these features indicate that Mg are very promising materials for implants

[Song et al., 2008, 2009].

The excellent ability of Mg alloys and Mg composites to be processed by die casting into

intricate shapes resulted in many applications in sports equipment and electronics devices.

They include the handles of archery bows, tennis rackets, golf clubs, bicycle frames (Fig.

1.1(e)), housings of cell phones and cameras, computers, laptops (Fig. 1.1(d)), and portable

media players.

Many more applications in structural components are envisaged for Mg in the future if

the difficulties associated with corrosion resistance and limited mechanical properties are

overcome. Although the mechanical properties of Mg alloys have been studied for many

decades, the body of knowledge on this material is much more limited than that for steel

or Al alloys. There is a lack of fundamental understanding of the key factors controlling

the macroscopic mechanical behavior of Mg and its alloys and this information is critical to

design novel alloys with improved microstructure. This situation is changing rapidly due

to the development of novel experimental techniques to characterize the microstructure

and the deformation mechanisms at the microscopic scale (electron backscatter diffraction,

X-ray computed tomography, in situ mechanical tests, etc.) and of advanced numerical

simulation tools (crystal plasticity, computational homogenization) that can provide a

detailed picture of the dominant deformation and fracture mechanisms in Mg as a function

of the loading conditions.

4

1.2 Deformation mechanisms of Mg alloys


Mg single crystals present a hexagonal closed packed (HCP) lattice, similar to that of

Be, Cd, Ti, Zn and Zr. The unit cell of the HCP lattice is a hexagonal prism which has two

hexagonal bases with sides of length a and height equal to c. Each vertex and the center

of these bases are occupied by one atom and a triangle of 3 atoms is also placed between

these 2 planes, see Fig. 1.2. The c/a ratio of Mg single crystals is 1.624, very close to the

theoretical value of 1.633 for contacting spheres.

Crystallographic features of HCP crystals, such as vectors and atomic plane families,

can be described using a four-value Miller index notation (hkil) in which the third index

i denotes a convenient but degenerate component which is the negative of the sum of the

first two (i = −h − k). The h, k and i index directions are separated by 120◦ and are

parallel to the axes a1, a2 and a3 in the basal plane of the prism. The l component is

perpendicular to the basal plane and parallel to the vertical axis c [Bravais, 1850] (Fig.

1.3 left side).

(a) (b)

Figure 1.2: HCP crystallographic structure

The elastic behavior of Mg single crystals presents transversely isotropic symmetry

due to the HCP lattice. The stiffness tensor C that relates the stress tensor S with the

elastic strain tensor Ee (S=CEe) can be characterized by means of the 5 independent

elastic constants C1111, C1122, C1133, C3333 and C1212, being direction 2 parallel to a2 and

direction 3 to c. The corresponding values are shown in Table 1.1 [Slutsky and Garland,

1957].

5


C1111 C1122 C1133 C3333 C1212

59.4 25.6 21.4 61.6 16.4

Table 1.1: Elastic constants (in GPa) of Mg single crystal at 300K [Slutsky and Garland,

1957].

Plastic deformation, contrary to elastic deformation, is irreversible and two main plastic

deformation mechanisms can be found in Mg, namely dislocation slip and twinning, Fig.

1.4(a). Plastic deformation by slip is due to the movement of dislocations in the atomic

planes with the highest atomic density and along the closed-packed orientations. They

correspond to the basal plane in Mg in three different orientations, namely h2110i, h1210iand h1120i. However, five independent slip systems are necessary to accommodate general

plastic deformation of the lattice [Taylor, 1938, Bishop and Hill, 1951] and basal slip can

only provide two. Thus, plastic deformation in other crystallographic planes with lower

atomic density is also necessary in HCP metals. In the case of Mg and Mg alloys, these

systems are, prismatic slip (f1010gh1210i) and pyramidal hai (f1011gh1210i) or hc+ai(f1010gh1210i), that are also represented on Fig. 1.3.

Figure 1.3: Plastic deformation modes in Mg

In addition to dislocation slip, plastic deformation in Mg (and other low symmetry

crystal structures) can occur by twinning, providing an additional mechanism to accom-

modate c-axis deformation. The availability of twinning deformation modes in HCP is

intimately tied to the c/a ratio [Yoo, 1981]. A mechanical twin formally corresponds to

a sheared volume for which the lattice orientation is transformed into its mirror image

across a so-called twin or habitus plane (oblique dividing plane defined by the twinning

6


direction, see Fig. 1.4 right). The sheared region of the crystal undergoes an irreversible

shear deformation of 0.129 [Zhang and Joshi, 2012]. Twins are easily observed by optical

microscopy as thin lines within the grains which divide the twinned region from the rest

of the crystal. The crystallographic orientation of the crystal within the twinned region is

different from that of the parent grain and this is readily observed by means of electron

backscatter diffraction (EBSD), as shown in Fig. 1.5. The development of twinning is a

process that involves two steps. The first one is the propagation of a thin twin band across

the grain, starting normally from the grain boundary. Afterwards, the twinned region

propagates in the direction perpendicular to the twin plane and eventually the twinned

region occupies most of the parent grain.

Two different twining modes have been reported in Mg, namely extension or tensile

twinning (f1012gh1011i) (the most commonly observed), that appears when the c axis

experiences tension, and contraction or compressive twinning (f2112gh2113i) (less com-

mon), that occurs under compression along the c axis [Reed-Hill and Robertson, 1957a,b,

Yoshinaga et al., 1973]. Contrary to plastic slip that may occur in either direction of the

slip vector, tension twinning only occurs in the direction that promotes the extension of

the c axis, while compression twinning takes place in the direction that leads to compres-

sion of the c axis. Thus, twinning –as opposed to dislocation slip – is a polar deformation

mechanism.

Figure 1.4: Permanent deformation after by slip and twinning.

7


Figure 1.5: EBSD image of Rh showing twins within the grains. [Kacher and Minor,

2014]

Plastic deformation in a given slip system is activated when the resolved shear stress, τα

reaches a critical value, the critical resolved shear stress (CRSS, ταc ), a material parameter

which depends on the chemistry, microstructure and deformation stage of the crystal. The

resolved shear stress on the system α (τα) is obtained by the projection of the stress

tensor S on the corresponding slip plane defined by its plane normal n and slip direction

s, according to

τα = S : s n (1.1)

In the case of a uniaxial loading, the plane normal (n) and the slip direction (s) are

given by the angles λ and ϕ (Fig. 1.6), and equation 1.1 is simplified to

τα = σ cos(λ) cos(ϕ) (1.2)

where m∗ = cos(λ) cos(ϕ) is the so-called Schmid factor.

Regarding twinning activation, the present trend follows the seminal developments by

Kalidindi [1998], Salem et al. [2005] and Staroselsky and Anand [1998] who included twin-

ning along with slip within the constitutive equation. This approach introduced twinning-

induced plasticity through a phenomenological evolution law for the twin volume fraction.

Twinning is modeled as a pseudo-slip mechanism and its activation is controlled by a CRSS

8


Figure 1.6: Geometric configuration to determine the resolved shear stress τα on the

slip system characterized by the normal plane n and the slip direction s under uniaxial

loading σapplied.

acting on the habitus twin plane and along the twinning direction and taking into account

the polar nature of twinning. In Mg alloys, the CRSS of compression twinning is 15 times

higher than that for tensile twining and, thus, very often only tensile twinning is activated

during deformation of Mg and Mg alloys [Zhang and Joshi, 2012].

In cubic materials (either FCC or BCC), general plastic deformation can be accom-

modated by one single slip system. In the case of Mg, the critical resolved shear stress

for the basal mode is much smaller than those of the other slip modes, but basal slip

can only provide two independent slip systems and cannot accommodate general plastic

deformation. This leads to the activation of twinning to accommodate the deformation

perpendicular to the basal plane, being basal slip and tensile twinning the most active

deformation modes in pure Mg and most Mg alloys because the CRSSs to active prismatic

or pyramidal slip are much higher. This trend may be affected, however, by the alloying

elements or temperature leading to changes in the most active modes.

Rolled Mg and Mg sheets present a marked basal texture and the c axis of the Mg

9


hexagonal crystals is aligned with the normal direction (ND) that is perpendicular to the

rolling direction (RD), see Fig. 1.7. The pole figure characteristic of this texture is depicted

in Fig. 1.7 left, where the accumulation of grains in ND shows the orientation of the c axis

along this direction.

Figure 1.7: (Left) Typical pole figure of rolled Mg along ND direction. (Right) Section

A-A corresponding with the plane defined by RD-ND axes. [Zhang and Joshi, 2012]

Under these conditions, the plastic deformation may be dramatically different depend-

ing on the loading direction, Fig. 1.8(a). This is depicted in Fig. 1.8(b), which shows the

tensile stress-strain curves of rolled AZ31 Mg alloy along different orientations with respect

to the normal direction, ND (from 0◦ to 90◦) [Liu et al., 2011]. Specimens tested along

an angle between 0◦ to 30◦ with respect to ND showed relatively lower yield strength due

to activation of extension twinning together with basal slip. In addition, the stress-strain

curves when twinning is active, present a particular “concave up” shape. This is due to the

progressive increases of twin volume fraction and to the final exhaustion when the most

part of the material has been transformed. When angles are larger than 60◦, basal slip and

pyramidal slip are the dominant deformation modes. The reason of that was that rolling

processes provokes that the vertical orientation of the crystals (parallel to c) is the ND,

favoring therefore the twinning activation when the tensile tests are performed in ND and

its inhibition when they are oriented with 90◦ to ND. [Jiang et al., 2008].

10

1.3 Modeling of polycrystal behavior

(a) (b)

Figure 1.8: (a) Orientation of the tensile axis with respect to the normal direction

ND. (b) Representative stress-strain curves of specimens tested at different angles with

respect to ND [Liu et al., 2011]


Structural components of metallic materials are made up of polycrystalline alloys. Poly-

crystal homogenization provides a bridge between micro and macroscale by means of inte-

gration of the microscopic strain and stress fields within the different grains to obtain the

macroscopic stresses and strains in the polycrystal. This kind of approach is applicable to

problems with a clear separation of scales, i.e. those in which the typical length-scale asso-

ciated with the gradients of the mechanical fields at the macroscale is large compared with

the typical length-scale of the polycrystalline microstructure (e.g. the grain or sub-grain

size).

Within this framework, the influence of the microscopic features of the polycrystal

(grain size, shape and orientations as well as elastic constants and the CRSS of the dif-

ferent slip and twinning modes) on the macroscopic response can be taken into account.

Polycrystal homogenization is a very complex, non-linear problem that has been solved with

two different approximations, namely mean-field methods [Taylor, 1938, Sachs, 1928, Moli-

nari et al., 1987, Lebensohn and Tome, 1993] and computational homogenization [Miehe

et al., 1999, 2002, Michel et al., 1999, Lebensohn et al., 2011, Segurado and Llorca, 2013].

Both of them rely on the definition of a Representative Volume Element (RVE) of the

microstructure, a crucial element to bridge micro and macroscales.

11


1.3.1 Representative volume element

The RVE is a sample of a heterogeneous material that fulfills the following conditions:

� It is entirely representative of the microstructure on average, and

� it is sufficiently large for the apparent properties to be independent of the surface

values of traction and displacement, so long as these values are macroscopically uniform

[Hill, 1963].

In essence, the first statement is about the material’s statistics (i.e. spatially homo-

geneous and ergodic), while the second one is a pronouncement on the independence of

effective constitutive response with respect to the applied boundary conditions. In the case

of polycrystals, the RVE is the smallest number of grains over which a measurement can

be made that will yield a value representative of the whole polycrystal. A simple periodic

unit cell is the RVE in the case of materials with periodic microstructure (Fig. 1.9(a)), but

the situation becomes much more complicated in random media, and 2D or 3D complex

cells which contain grains with different sizes, shapes and orientations are necessary (Fig.

1.9(b)).

Very accurate data can be currently obtained of the grain size, shape and orientation

in polycrystals owing to the development of advanced 3D microstructural characterization

techniques (such as serial sectioning and X-ray microtomography together with electron

back-scattered diffraction and X-ray diffraction) [Ludwig et al., 2009, Robertson et al.,

2011, Fernandez et al., 2013, Sket et al., 2014]. Grain size and shape statistical functions,

together with the orientation distribution function (that characterizes the texture) can be

used by means of Monte Carlo lotteries to build up RVEs of the polycrystal microstructure.

The second key ingredient to simulate the polycrystal behavior is the complex behavior

of the single crystals, which should include both plastic deformation by slip and twinning

in the case of Mg. The framework for this task is the well established crystal plasticity

theory [Kroner, 1961, Mandel, 1972, Asaro and Rice, 1977], to describe the homogeneous

and heterogeneous deformation and hardening of single crystals under complex loading

conditions.

12


(a)

(b)

Figure 1.9: Periodic microstructure and the corresponding RVE (a). Random poly-

crystal microstructure and the corresponding RVE (taken from [Segurado and Llorca,

2013]).

1.3.2 Crystal plasticity model

Crystal plasticity estimates the plastic deformation that undergoes a single crystal

under certain boundary conditions. Because plastic deformation, specially under forming

process, can be substantially large, the kinematics of crystal deformation under finite

strains should be established previously.

A region in the three-dimensional space R3 is assigned to the material body B. The

points within this region are called particles or material points. Different configurations

or states of the body correspond to different regions in the 3D space. B0 and B are the

undeformed and deformed configuration at times t0 and t, respectively (Fig. 1.10). The

positions of the material points in the undeformed (or reference) configuration are given

by vector x, whereas those in the deformed (or current) configuration are denoted by y.

13


Thus, the displacement in the deformed configuration is given by u = y � x.

Figure 1.10: Reference or undeformed configuration (B0) and current or deformed

configuration (B). Notation.

The deformation dy of a material line segment dx at x in the reference configuration

is given by means of the deformation gradient tensor F as follows

dy =∂y

∂xdx = Fdx (1.3)

The velocity of the material point x is given by

v =d

dtu = u (1.4)

and the velocity gradient L, which expressed the relative velocity between two positions in

the deformed configuration, can be expressed as function of deformation gradient F as,

L =∂v

∂y= FF−1 (1.5)

The elasto-plastic deformation of the single crystal is accounted for by means of the

multiplicative decomposition [Kroner, 1961]. The single crystal deformation can be decom-

posed into two components Fe and Fp, see Fig. 2.1. The elastic deformation gradient, Fe,

includes the recoverable distortion of the lattice as well as the rigid-body rotations while

Fp accounts for the irreversible plastic deformation induced by plastic slip and twinning.

In this sense, transformation of the reference state by Fp leads to an intermediate config-

uration, Bint, corresponding to a fictitious state of the body in which each material point

14


is unloaded and with its particular lattice coordinate system coinciding with the system in

which the constitutive equations are written.

Figure 1.11: Multiplicative decomposition of the total deformation gradient F into the

elastic, Fe, and plastic, Fp, components.

The transformation from the reference configuration to this intermediate configuration

hence needs to include the flow of material expressed in the constant lattice frame. The

subsequent transformation from the intermediate to the current configuration, correspond-

ing to elastic stretching of the lattice (plus rigid-body rotations), is characterized by the

elastic deformation Fe. Therefore, the overall deformation gradient relating the reference

to the current configuration follows from the sequence of both contributions as

F = FeFp (1.6)

The evolution of the plastic deformation gradient Fp can be expressed as function of

velocity gradient Lp, following the definition 1.5 applied to Fp, leading to

Fp = LpFp (1.7)

and it can be expressed as [Rice, 1971],

Lp =N∑α=1

γαsα nα (1.8)

15


if plastic deformation takes place by dislocation slip. The vectors sα and nα stand, re-

spectively, for unit vectors in the slip direction and the normal to the slip plane of the

slip system α and N is the number of slip systems. The term γα is the shear rate for the

system α which is a function of the resolved shear stress, τα, and the critical resolved shear

stress, ταc

γα = f(τα, ταc ) (1.9)

with

τα = S : (sα nα) (1.10)

ταc = g(γ, γ) (1.11)

where S is the second Piola-Kirchhoff stress tensor and γ and γ stand for the total shear

strain on each system and the shear strain rate, respectively. Equations 1.8,1.9 and 1.11

will be reviewed in more detail in Chapter 3, where this model will be particularized for

Mg alloys.

1.3.3 Mean-field approximations

Both mean-field approximations and computational homogenization are built upon the

assumption of separation of scales illustrated in Fig. 1.12. The constitutive response of ma-

terial in the macroscale is obtained by solving a boundary value problem in a representative

volume element of this microstructure given by the subdomain β0.

The macroscopic (or effective) constitutive equation is given by the relation between S

(the effective first Piola-Kirchhoff stress tensor) and F (the effective deformation gradient

tensor). They can be expressed as

F =1

V0

∫β0

F(x)dV0 (1.12)

S =1

V0

∫β0

S(x)dV0 (1.13)

The mean-field approximation considers that the microfields in each grain can be repre-

sented by a single value, that is the volume-average of the corresponding microfield inside

the crystal. Usually, the microstructure defined in the subdomain β0 is made then by a

set of M inclusions βi inside a matrix, whose size, shape and orientation correspond to the

16


Polycrystalline microstructure at material point 𝒙

Macroscale Sample

𝑥

Computational subdomain 𝜷𝟎

Mean field subdomain 𝜷𝟎

Figure 1.12: Separation of scales between microscale and macroscale.

single crystals in the polycrystal. The effective stress and strain deformation tensors can

be expressed as

F =1

V0

M∑i

∫βi

FdVi =1

V0

M∑i

VihFii (1.14)

S =1

V0

M∑i

∫βi

SdV0 =1

V0

M∑i

VihSii (1.15)

where hFii and hSii stand for the volume-averaged deformation gradient and stress tensor,

respectively, in inclusion i and Vi for the volume of inclusion i in the subdomain.

The different mean-field approximations adopt different hypothesis for the magnitude

of hFii or hSii. The most simple ones are the isostrain (hFii = F) or isostress approaches

(hSii = S). The first one assumes that all the inclusions undergo the same deforma-

tion while the second one proposes that the stress carried by all inclusions is equivalent.

Both models were developed, respectively, by Taylor [1938] and Sachs [1928]. They are

based on assumptions that disregard the shape and local neighborhood of the inclusions

17


and generally violate equilibrium and compatibility conditions, respectively. These models

may provide relatively accurate approximations of the polycrystal behavior if the single

crystals are almost isotropic and posses a large number of slip systems to accommodate the

deformation (FCC and BCC materials), but fail if there are large differences in the strains

or stresses carried by individual grains, as it turns out to be the case in HCP crystals.

Furthermore, although the isostrain approach fulfills the compatibility condition, leads to

a very stiff response. More accurate models were developed in the context of Eshelby’s

approach [Eshelby, 1957] and of particular linearization schemes to obtain the polycrystal

behavior. Among them, the viscoplastic self-consistent scheme (VPSC) has become the

standard tool to homogenize the plastic deformation of polycrystals. This formulation,

based on a ad-hoc linearization of the non-linear single crystal constitutive behavior and

on the use of the linear self-consistent approximation, was first proposed by Molinari et al.

[Molinari et al., 1987] to predict the texture evolution of polycrystalline materials, and it

was later extended and implemented numerically by Lebensohn and Tome [Lebensohn and

Tome, 1993] in the so-called VPSC code. The main features of the VPSC strategy will be

briefly reviewed below.

The VPSC model assumes that the interaction of a grain with the surrounding matrix

can be approximated by the interaction between the grain and a hypothetical homogeneous

medium (HEM), which is characterized by an average constitutive behavior of the entire

polycrystal aggregate. Each grain corresponds to a particular orientation of the ODF and

its volume fraction is taken as the weight of that particular orientation in the ODF. The

grains are represented as ellipsoidal inclusions, Fig. 1.13.2. Implementation of VPSC as FE material model

+ …

+

HEMgrain

≈

inclusion problem→ Eshelby solution: linear !

'pxσ'pxσ

Gran:

op

appVPSCVPSCp :M ε+σ=ε &&HEM:

linearization !??

σ∂

ε∂==

VPSCpggVPSC B:MM&

localization tensors: f (Mg,MVPSC,Eshelby tensor)

ogp

ggop b:M ε+=ε &&

Self-consistent equations:

(also: )

Figure 1.13: VPSC assupmtion where the matrix-grain interaction is approximated by

a ellipsoidal grain (with its particular orientation) within a HEM

18


In contrast to Taylor or Sachs approaches, the relation between the crystal microfields (

σ′c and εc) and the average polycrystal macroscopic fields ( σ′px εpx ) in VPSC, is different

for each crystal and depends on the particular orientation of the crystal with respect to

the HEM, Fig. 1.13.

The standard version of VPSC is rigid-viscoplastic, an elastic stresses are neglected at

both macroscopic polycrystalline and grain levels. Following this assumption, the macro-

scopic or polycrystalline deviatoric strain rate tensor εpx is related to a macroscopic devi-

atoric stress tensor σ′px through a non-linear viscous relation. This non-linear relation is

linearized at a given stress by

εpx = Mpx : σ′px + εpx0 (1.16)

where Mpx and εpx0 stand for the tangent viscoplastic compliance and the back-extrapolated

strain rate, respectively. On the microscale, following the mean field assumptions, the

behavior of each crystal (or orientation) c is solely represented by its average fields εc and

σ′c. The constitutive relation assumed for the whole grain is a power-law viscoplastic

relation given by,

εc = γ0

N∑α=1

(σ′c : (sα(c) nα(c))Sym

ταc

)n(sα(c) nα(c))Sym (1.17)

where sα(c) and nα(c) are the tangent and normal vectors of the system α of grain c,

(sα(c) nα(c))Sym is the symmetric Schmid tensor, ταc is the CRSS of system α in grain c

and γ0 and n stand for the reference strain rate and rate sensitivity exponent, respectively.

This viscous relation in eq 1.17 for each grain c is also linearized as,

εc = Mc : σ′c + εc0 (1.18)

where Mc and εc0 stand for the tangent viscoplastic compliance and back extrapolated

strain rate of grain c.

The localization equations in a mean-field model provide the relationship between mi-

crofields and macrofields. In the VPSC approach, the localization stress tensors Bc and bc

can be written as

σ′c = Bc(Mc,Mpx,S)σ′px + bc(Mc,Mpx,S, ε0, εc0) (1.19)

where S is the Eshelby’s tensor. The Eshelby tensor S stands for an anisotropic ellipsoidal

inclusion embedded in an anisotropic media and, contrary to the isotropic case, analytical

19


expressions are not available. Thus, it has to be computed numerically for each orientation

using Green functions. The particular expressions for the localization tensors Bc and bc

can be found in the literature [Segurado et al., 2012, Lebensohn and Tome, 1993] and are

not given here for brevity.

Polycrystalline fields can be obtained as an average over the crystal fields. For instance,

in the case of strain rates,

εpx =< εc > (1.20)

Finally, combining expression 1.16, 1.18, 1.19 and 1.20, the following self consistent

equations are obtained

Mpx =<Mc : Bc > (1.21)

εpx0 =<Mc : bc + εc0 > (1.22)

This implicit set of equations can be solved iteratively to obtain Mpx and εpx0 . The VPSC

model is used to simulate the polycrystalline response and microfield evolution under a

given strain or stress history. This history is discretized in increments to obtain both the

macroscopic polycrystalline behavior and the microscopic (grain) fields.

1.3.4 Computational homogenization

Mean-field models (and, particularly, the VPSC approximation) have demonstrated

their ability to predict the average flow stress and the texture evolution in polycrystals

and they have been recently used to provide constitutive equations for these materials

within the context of multiscale simulations [Segurado et al., 2012]. However, these models

cannot capture the local stress and strain fields accurately (they generally use only a mean

value to represent the distribution of fields inside the grain) and this may lead to large

differences at the local level for highly anisotropic crystals. In addition, the statistical

treatment of the microstructure does not allow to analyze the influence of the actual grain

shape and local details of the grain spatial distribution (i.e. clusters of second phases or

grain orientations, etc). Under these circumstances, more sophisticated models based on

computational homogenization have to be used to capture these local effects.

Computational homogenization is based on the numerical simulation of the mechanical

behavior of a representative volume element (RVE) of the material microstructure. The

numerical solution of the boundary problem is carried out using different techniques, which

20

1.4 Mechanical behavior of single crystals

include the Fast Fourier Transform method [Michel et al., 1999], recently extended to

viscoplastic polycrystals [Lebensohn et al., 2011], and the finite element method [Miehe

et al., 1999, 2002]).

Three different types of discretization of the RVE can be carried out. The first one

is a voxel-based model in which the RVE is made up by a regular mesh of N � N � Ncubic elements, Fig. 1.14(a). Each cubic element stands for a single crystalline grain and

thus the model can include a large number of grains. While this is important from the

statistical viewpoint, this representation of the microstructure leads to a poor description of

the grain shape and of the strain fields within the grains. Another possibility to represent

the microstructure is depicted in Fig. 1.14(b). The discretization is also carried out

with cubic elements but each crystal was represented with many elements and, thus, the

model includes information about the distribution of grain sizes and shapes within the

polycrystal. In addition, complex deformation fields can be accounted for within each

grain. Nevertheless, the jagged shape of the grain boundaries is not realistic and this leads

to a third type or representation (Fig. 1.14(c)), in which each grain is a polyhedron which

is obtained by means of a Voronoi tessellation. Each polyhedron is discretized with a finite

element mesh to capture the stress and strain gradients within the crystal. This third

representation of the microstructure is obviously more realistic but the higher cost (from

the viewpoint of the generation of the microstructure and of the computational resources)

is not always associated with a dramatic improvement in the accuracy of the predictions

and the RVE in Fig. 1.14(b) is often preferred.

From the viewpoint of the boundary conditions, it is nowadays well established that the

best results are obtained if periodic boundary conditions are applied to the RVE [Segurado

and Llorca, 2002] because the effective behavior derived under these conditions is always

closer to the exact solution (obtained for an RVE of infinite size) than those obtained under

imposed displacements or forces (Huet [1990], Hazanov and Huet [1994]). Further details

about the periodic boundary conditions are explained in section 2.3.


The physical deformation mechanisms in metallic single crystals have been studied in

detail and they are well understood. The elastic behavior is determined by the crystal sym-

metry and the corresponding elastic constants, which are well known. Plastic deformation

21


(a) (b) (c)

Polycrystalline homogenization• Polycrystal behavior is obtained by FEM analysis of a RVE of the

microstructure• Three type of periodic RVEs are considered:

• The grain orientations are generated by MC to be statisticallyrepresentative of ODF

• The microstructures of (b)3 and (c) are synthetically obtained to fitstatistics on grain sizes and shapes

• Periodic boundary conditions are used and load history is introducedby 9 independent terms of F(t) and Σ(t)

3Dream3D

Figure 1.14: Discretization of RVE of polycrystals. (a) Model with 1000 cubic voxels, in

which each one stands for a single crystal. (b) Model containing 100 crystals discretized

with 64000 voxels. (c) Model in which each crystal is represented by a polyhedron

obtained by means of a Voronoi tessellation.

is controlled by dislocation slip and, in some cases, by twinning and it can be highly depen-

dent on the crystal orientation, leading to a strong anisotropy in the plastic response. The

single crystal behavior is modeled within the continuum viewpoint with crystal plasticity

models [Hill, 1966, Rice, 1971, Hill and Rice, 1972], which take into account the geometry

of slip and/or twinning for each material and lattice configuration, see section 1.3.2. The

response of each slip/twinning system is governed by the critical resolved shear stresses

(CRSS) and its evolution with deformation is introduced by means of either phenomeno-

logical [Asaro and Needleman, 1985, Bassani and Wu, 1991] or physically-based models

[Arsenlis and Parks, 2002, Cheong and Busso, 2004, Ma et al., 2006]. Thus, although the

theoretical framework to simulate the mechanical behavior of single crystals is available,

quantitative values of the parameters in these models are difficult to obtain experimentally,

limiting the predictive capabilities of the polycrystal homogenization.

There are three different approaches available to obtain the quantitative values of the

parameters which control the single crystal behavior. The first one is to carry out simple

mechanical tests of microscopic single crystals built from the polycrystal (see Gianola and

Eberl [2009] for a review) by means of focus ion beam milling. The microscopic single

crystals have often a circular section with a diameter in the range 1 to 10 µm and can be

tested in compression with a flat punch in a nanoindenter. By choosing the orientation of

the parent grain, compression tests can be carried out in particular orientations to activate

only one slip system and thus to obtain the CRSS as well as the strain hardening of each

22


slip system. However, this is particularly difficult in single crystals which present a strong

plastic anisotropy (e.g. Mg) because deformation tend to be dominated by softest slip

modes regardless of the initial orientation of the crystal [Prasad et al., 2014, Ye et al., 2011,

Kim, 2011] (Fig. 1.15). Moreover, the quantitative values of the CRSS and of the strain

hardening for each slip system cannot be directly used in the simulation of polycrystals

because of the presence of size effects.

Figure 1.15: Mg micropillar after compression in a direction at 45◦ from the basal plane

normal, showing slip along the basal plane. Courtesy of Yuan-Wei Edward Chang

An alternative strategy, experimentally less challenging, is based on the use of instru-

mented nanoindentation of single crystals with different orientation within the polycrystal

[Liu et al., 2005, Eidel, 2011, Sanchez-Martın et al., 2014]. Testing is very straight forward

in this case but the interpretation of the experimental data to obtain the parameters which

control the behavior of each slip/twinning system is difficult due to the complex stress state

below the indenter and. In addition, nanoindentation results are also size dependent.

Another methodology to obtain the single crystal properties is based on a multsicale

modeling approach. In this case, the effect of alloying elements, precipitates or defects

and dislocation-dislocation interactions on the CRSS and the subsequent hardening are

accounted for using density-functional theory, molecular dynamics or dislocation dynam-

ics. Successful examples of this methodology have appeared recently [Leyson et al., 2010,

Barton et al., 2013] but they are still limited in terms of the mechanisms that can be

accounted for and of the uncertainties associated with the bridge of time and length scales

23


between the different simulation approaches.

Thus, taking into account the limitations of experiments and theory, the most widely

used strategy to obtain the single crystal properties is based on the calibration of the

parameters which control the single crystal properties by fitting experimental results of

polycrystals loaded in different orientations by means of simulations based on mean-field

methods or computational homogenization . The main problem with this strategy is that

the number of parameters to be determined for each single crystal is very large and finding

the optimum parameter set is neither easy nor a unique result is guaranteed. In fact, it is

not unusual to find that different authors report different (or even contradictory) values for

similar materials. HCP metals are the most typical example of these shortcomings because

of the large plastic anisotropy and the coexistence of slip and twinning during plastic

deformation. For instance, Table 1.2 shows the magnitude of the initial CRSS reported by

different groups for the most important slip modes (basal, prismatic and pyramidalhc+ai)and extension twinning in AZ31 Mg alloy. The differences are non negligible from the

quantitative viewpoint and, in addition, some authors [Agnew et al., 2001, Liu et al.,

2011] considered that the initial CRSS for tensile twinning was below the one for basal slip

whereas basal was the softest mode in other studies [Fernandez et al., 2011, Knezevic et al.,

2010, Wang et al., 2010], following the behavior of pure Mg. Obviously, these differences

have very large implications in the dominant deformation mechanisms (and in the texture

development) during deformation and their origin is not easy to assess. Although disparities

in grain size or processing parameters could explain some of the differences in the initial

CRSS reported on the different studies, the spread in the corresponding experimental

results is much smaller than the differences among the CRSS values. This fact suggests

that the disparities in the values proposed for the CRSS should also be closely related to

the methodology used for the model calibration.

1.5 Objectives and structure of the thesis

Polycrystal homogenization is a powerful tool to obtain the mechanical properties of

polycrystalline alloys that relies in three ingredients: an accurate representation of the

microstructure (included in the RVE), a robust homogenization strategy (either based on

mean-field or computational methods) and accurate information about the single crystal

mechanical properties within the polycrystal. A huge progress has been achieved in the

24

1.5 Objectives and structure of the thesis

Deformation mode reference

Fernandez et al. Liu et al. Knezevic et al. Wang et al. Agnew et al.

Basal α α α α α

Prismatic 9α 2α 5α 5α —

Pyramidalhc+ai 13α 15α 6α 8α 3α

Twinning 2α 0.7α 2α 2α 0.5α

α (MPa) 9 — — 15 30

grain size (µm) 13 42 8 — 25-100

Table 1.2: Values of the initial CRSS for different slips modes and tensile twinning in

AZ31 Mg alloy predicted by fitting experimental results on polycrystals with simulations

based on mean-field methods or computational homogenization.

first two areas in the last decades and the Achilles’ heel of polycrystal homogenization

is the lack of a robust methodology (either experimental, theoretical or mixed) to obtain

accurate, quantitative values for the mechanical properties of the single crystal, including

the CRSS of the different slip/twinning modes and the corresponding strain hardening

rates.

The standard approach to obtain this information is based in inverse analysis in which

the single crystal properties are obtained by fitting the predictions the polycrystal homog-

enization model for different loading conditions to experimental results. This is normally

carried out by a trial and error approach and the accuracy of the resulting parameters is

often uncertain because the problem is highly nonlinear, the number of parameters to be

determined for each single crystal is very large and a unique result is not always guaranteed.

The main objective of this thesis is to develop a robust and reliable inverse optimiza-

tion methodology to obtain the single crystal properties from the mechanical behavior of

polycrystals, which can be applied to strongly anisotropic HCP metals deforming by slip

and twinning. The polycrystal behavior will be obtained by means of the finite element

simulation of an RVE of the microstructure and the inverse problem will be solved by

means of the Levenberg-Marquardt method [Levenberg, 1944, Marquardt, 1963], which

is recommended for general non-linear least squares problems in optimization literature

[Dennis and Schnabel, 1996]. The robustness and accuracy of the methodology will be

assessed by comparing the predictions provided by computational homogenization with

independent experimental results. In addition, the influence of the input information on

25


the accuracy of the results will be studied.

This methodology will be applied to two Mg alloys of large technological interest.

Firstly, heavily textured rolled AZ31 Mg sheets, whose mechanical behavior is strongly de-

pendent on the orientation with respect to the rolling direction, will be analyzed. Secondly,

MN10 and MN11 Mg alloys will be studied. These are rare earth-containing alloys which

present a weaker texture and more limited differences among the CRSS of the different slip

modes.

To fulfill these objectives, the thesis is structured as follows. After the introduction,

the second chapter presents the models and algorithms developed to perform the numerical

simulation of Mg and its alloys. This chapter is structured in three sections. The first one

is devoted to the crystal plasticity model adapted for Mg alloys. The second section

presents the computational homogenization strategy for polycrystalline Mg alloys and the

inverse optimization methodology is detailed in section 3. The next chapter presents the

application of this methodology to Mg alloys and also includes the analysis of the robustness

of the approach. Finally, the conclusions and the future work are summarized in chapter

4.

26

Chapter 2Models and algorithms

2.1 Finite element crystal plasticity model

The mechanical behavior of polycrystalline Magnesium alloys can be predicted using

homogenization models that provide the macroscopic response as function of the crystal

behavior and the polycrystalline microstructure (grain size, shape and orientation dis-

tributions). In addition to the use of an appropriate homogenization technique (either

mean-field models or computational homogenization), three elements are fundamental for

an accurate prediction of the behavior of the polycrystal: (1) A constitutive model for

the behavior of the grains that reproduces the actual deformation mechanisms of the crys-

tal, (2) a realistic and representative description of the microstructure and (3), a set of

parameters that accurately describe the deformation of grains using previous model.

In this chapter, the models and algorithms developed to create a computational homoge-

nization framework for predicting the behavior of Magnesium alloys will be described. With

respect to the crystal behavior (1), the general crystal plasticity (CP) framework will be

presented together with the description of the particular CP model developed for Mg and

its numerical implementation in the finite element context. Next, the microstructure repre-

sentation (2) and the computational homogenization technique will be presented. Finally,

the development of an inverse optimization technique to obtain the crystal parameters of

a Mg alloy (3) from actual microstructure and macroscopic tests will be described.

27

Chapter 2. Models and algorithms

2.2 Crystal plasticity model for Mg alloys

A crystal plasticity model has been developed and implemented as a user material

subroutine (UMAT) in the finite element code ABAQUS [Abaqus, 2013]. The UMAT

developed here for Mg alloys is based on the subroutine developed and implemented pre-

viously for Titanium [Segurado and Llorca, 2013]. The original model was able to account

for crystals with different lattices (FCC, HCP, BCC) and several types of hardening laws

but the only plastic deformation mechanism accounted for was dislocation slip. However,

an accurate description of the crystalline deformation in Mg alloys should undoubtedly

include twinning deformation. For this reason, the original model [Segurado and Llorca,

2013] has been enhanced to simulate the behavior of Mg alloys by including a model for

twinning deformation and other particular issues as non-Schmid effects on CRSS.

The crystal plasticity formulation proposed here is based on the multiplicative decom-

position of the deformation gradient in its elastic and plastic parts, according to

F = FeFp (2.1)

The total velocity gradient L (eq. 1.5 in section 1.3.2) can then be expressed as

L = FF−1 = FeFe−1

+ FeFpFp−1

Fe−1

(2.2)

where Lp = FpFp−1stands for the plastic velocity gradient in the intermediate or relaxed

configuration.

The plastic deformation is accommodated by two deformation mechanisms, slip and

twin, being Nsl and Ntw the total number of slip and twinning systems available, respec-

tively. Twinning is included in the crystal plasticity framework using the model developed

by Kalidindi [Kalidindi, 1998]. A material point is divided into two phases, a parent region

and a twinned region (Fig. 2.1), which is formed by a maximum of Ntw subregions. Each

subregion belongs to a given twinning system α and its volume fraction is fα. Thus, the

parent region volume fraction is given by 1�∑Ntw

α=1 fα.

Under this approach the material point can be considered as a composite material in

which the iso-strain hypothesis holds (F and Fe are the same in all phases). The plastic

deformation is the result of three mechanisms and the plastic velocity gradient in the

intermediate configuration contains three terms, related with the slip, twinning and re-slip

mechanisms, Lpsl, Lp

tw, and Lpre−sl respectively.

28


Figure 2.1: Multiplicative decomposition indicating material point subdivision in parent

and twin phases

Lp = Lpsl + Lp

tw + Lpre−sl (2.3)

The slip in the parent phase, Lpsl, is given by

Lpsl =

(1�

Ntw∑α=1

fα) Nsl∑

i=1

γisisl nisl (2.4)

where sisl and nisl stand, respectively, for the unit vectors in the slip and normal direction

to the slip plane considered in the intermediate configuration.

The second contribution, Lptw, is the rate of deformation due to the twin transformation

of a differential volume fraction of parent phase dfα

Lptw =

Ntw∑α=1

fαγtwsαtw nαtw (2.5)

where fα = dfα/dt is the rate of the volume fraction transformation in the twin system α,

sαtw and nαtw are the unit vectors defining the twinning system and γtw is the characteristic

shear of the twinning mode (in the case of tension twinning of Mg alloys, γtw =0.129,

[Zhang and Joshi, 2012]). It is recalled that extension twinning is a polar mechanism and

it will only take place when the applied deformation leads to extension of the c axis of the

HCP lattice.

Finally, the third contribution corresponds to the slip of the transformed regions (here

denominated as re-slip), Lpre−sl, which can be expressed as,

29


Lpre−sl =

Ntw∑α=1

fα

(Nsl−tw∑i∗=1

γi∗si

∗

sl ni∗

sl

)(2.6)

where si∗sl and ni∗

sl stand for the unit vectors in the slip and normal directions to the slip

system i considered and re-oriented due to the twinning transformation of that region. The

reorientation is defined by a rotation tensor Qα

Qα = 2nαtw nαtw � I (2.7)

where I is the second order identity tensor.

It has been experimentally observed that the volume fraction of twinned regions in

many Mg alloy [Fernandez et al., 2013, Kalidindi, 1998, Remy, 1981] reaches a maximum

around∑fα � 0.80. Thus, the re-slip term is activated at a given material point when the

volume fraction of the twinned material at this point reaches 0.80. The number of systems

considered for re-slip, Nsl−tw, might be smaller than the number of original slip systems

Nsl for computational efficiency. Then, the total number of re-slip systems (Nre−slip) will

be obtained by the product of the number of slip systems that can undergo re-slip (Nsl−tw),

and the number of twinning systems (Ntw), that is:

Nre−slip = NtwNsl−tw (2.8)

The crystal was assumed to behave as an elasto-viscoplastic solid in which the plastic

slip rate for a given slip system follows a power-law, according to [Hutchinson, 1976],

γi = γ0

(jτ ijτ ic

) 1m

sign(τ i) (2.9)

where γ0 is a reference shear strain rate, τ ic the CRSS of the slip system i, m the rate-

sensitivity exponent and τ i the resolved shear stress on the slip system i.

Similarly, the twinning rate on the twinning system α, fα, also follows a viscous law

fα = f0

(hταiταc

) 1m

with hτi =

{τ if τ � 0

0 if τ < 0(2.10)

and the transformation rate is set equal to zero if the volume fraction of twinned material

exceeds a saturation value of 0.80 [Kalidindi, 1998]. Mathematically,

30


fα = 0 ifNtw∑α=1

fα � 0.80 (2.11)

Because of the iso-strain approach, the parent and twinned phases at a given material

point are deformed under the same F and Fe and they share the same elastic strain in the

intermediate configuration, given here by the Green-Lagrange strain tensor, Ee,

Ee =1

2

(FeT Fe � I

). (2.12)

The symmetric second Piola-Kirchhoff stress tensor in the intermediate configuration,

S, is obtained in this case from the volume-averaged stress tensors in the different phases

S =

(1�

Ntw∑α=1

fα)

Sparent +Ntw∑α=1

fαSα (2.13)

and the stresses on the parent (Sparent) and twinned (Sα) phases are given by

Sparent = CEe

Sα = CαEe (2.14)

where C stands for the fourth order elastic stiffness tensor of the crystal in its original

orientation and Cα are the corresponding stiffness tensors reoriented after twinning. They

are given by,

Cαijkl = Cα

pqrsQαipQ

αjqQ

αkrQ

αls (2.15)

The resolved shear stress on a slip (τ i) or twinning (τα) system in the parent (i) region

is obtained as,

τ i = Sparent : sisl nisl with τα = Sparent : sαtw nαtw (2.16)

while the resolved shear stress on a slip system in the twinned region (τ i∗) is given by,

τ i∗

= Sα : si∗

sl ni∗

sl (2.17)

Finally, the Cauchy stress can be approximated as

31


σ = J−1FeSFeT � ReSReT (2.18)

under the assumption of small elastic deformations, where J = det(F) and Re stands for

the orthogonal rotation tensor obtained by the polar decomposition of Fe.

The last ingredient of the model consists on the evolution equations of the CRSSs

of each system (terms τc in equations 2.9 and 2.10). The initial values (in absence of

previous plastic deformation) of the CRSSs are given by τ i0,c or τα0,c for a slip system i and

a twin system α, respectively. A phenomenological hardening model is considered for the

evolution of the CRSSs, which is able to reproduce the different stages of single crystal

deformation [Kothari and Anand, 1998]. The evolution of the CRSS τ ic , ταc , τ i∗c for slip,

twin and re-slip systems are then given by equations 2.19, 2.20 and 2.21, respectively,

τ ic = qsl−sl

Nsl∑j=1

h0j

(1� τ j

τ jsat

)asljγjj+ qtw−sl

Ntw∑β=1

h0tw

(1� τβ

τ twsat

)atwjγβj (2.19)

ταc = qtw−tw

Ntw∑β=1

h0tw

(1� τβ

τ twsat

)atwfαγtw (2.20)

τ i∗

c = qsl−sl

Nre−sl∑j=1

h0j

(1� τ j

τ jsat

)asljγjj (2.21)

where the different parameters in these equations define the contributions arising from self

hardening and latent hardening. The self hardening of a given slip (i) or twinning (α)

system correspond to the evolution of the CRSS only due to plastic deformation on that

particular system. This evolution is defined by three terms: the saturation stress, τsat, the

initial hardening rate h0 and the hardening exponent a. The evolution of the CRSS (τc) in

a given system isolated is depicted in Fig. 2.2. This figure shows how the initial value of

CRSS (τo,c) evolves, with a initial tangent modulus of h0, until it reaches the value of τsat.

The latent-hardening contribution to slip due to slip in other systems is introduced

with the coefficient qsl−sl whereas the contribution induced by twinning is given by qtw−sl.

The model only takes into account the effect of twinning on slip and it is assumed that slip

does not influence twinning (qsl−tw = 0) [Capolungo et al., 2009, Zhang and Joshi, 2012].

Three slip modes (basal, prismatic and first pyramidal hc+ai) and tensile twinning have

been included in the model to simulate the deformation of AZ31, MN10 and MN11 Mg

32


Figure 2.2: CRSS evolution by hardening

alloys. These deformation modes are depicted in Fig. 1.3 on page 6, and defined according

to its normal plane n (red line) and slip direction s (blue line). Each deformation mode

comprises several slip/twin systems considering the lattice symmetry of an hexagonal cell

and the resulting set of 24 systems are shown in Table 2.1, both expressed in Bravais

notation (a1,a2,a3,c) and in an orthogonal reference system, more useful for the numerical

implementation. The orthogonal system is defined by three vectors e1, e2, e3, fixed to the

hexagonal cell and defined as e1=a2�c, e2=a2 and e3=c, Fig. 2.3.

Figure 2.3: Different reference systems used to characterize the planes in the hexagonal

lattice. Bravais (left) and orthogonal reference system (right)

2.2.1 Time discretization

The non-linear global FE problem is solved by applying the boundary conditions (loads

or displacements) as a function of a time. The solution is obtained by imposing global

33


Slip mode Slip syst nbrav sbrav nort sort

Basal 1 0 0 0 1 2 -1 -1 0 0 0 1 -0.8661 0.5 0

Basal 2 0 0 0 1 -1 2 -1 0 0 0 1 -0.8661 -0.5 0

Basal 3 0 0 0 1 -1 -1 2 0 0 0 1 0 1 0

Prismatic 1 1 0 -1 0 -1 2 -1 0 1 0 0 0 1 0

Prismatic 2 0 -1 1 0 2 -1 -1 0 0.5 0.8661 0 -0.8661 0.5 0

Prismatic 3 -1 1 0 0 -1 -1 2 0 0.5 -0.8661 0 -0.8661 -0.5 0

Pyrhc+ ai 1 1 0 -1 1 -1 -1 2 3 0.8823 0 0.4708 -0.4543 0.2623 0.8514

Pyrhc+ ai 2 1 0 -1 1 -2 1 1 3 0.8823 0 0.4708 -0.4543 -0.2623 0.8514

Pyrhc+ ai 3 0 -1 1 1 1 1 -2 3 0.4411 0.7641 0.4708 -0.4543 -0.2623 0.8514

Pyrhc+ ai 4 0 -1 1 1 -1 2 -1 3 0.4411 0.7641 0.4708 0 -0.5246 0.8514

Pyrhc+ ai 5 -1 1 0 1 2 -1 -1 3 -0.4411 0.7641 0.4708 0 0.5246 -0.8514

Pyrhc+ ai 6 -1 1 0 1 1 -2 1 3 -0.4411 0.7641 0.4708 0.4543 -0.2623 0.8514

Pyrhc+ ai 7 -1 0 1 1 2 -1 -1 3 0.8823 0 -0.4708 0.4543 0.2623 0.8514

Pyrhc+ ai 8 -1 0 1 1 1 1 -2 3 0.8823 0 -0.4708 0.4543 -0.2623 0.8514

Pyrhc+ ai 9 0 1 -1 1 -1 -1 2 3 0.4411 0.7641 -0.4708 0 0.5246 0.8514

Pyrhc+ ai 10 0 1 -1 1 1 -2 1 3 0.4411 0.7641 -0.4708 0.4543 0.2623 0.8514

Pyrhc+ ai 11 1 -1 0 1 -2 1 1 3 -0.4411 0.7641 -0.4708 0 0.5246 0.8514

Pyrhc+ ai 12 1 -1 0 1 -1 2 -1 3 -0.4411 0.7641 -0.4708 -0.4543 0.2623 0.8514

Tensile Twin 1 1 0 -1 2 -1 0 1 1 0.6838 0 0.7298 -0.7298 0 0.6838

Tensile Twin 2 0 1 -1 2 0 -1 1 1 0.3419 0.5922 0.7298 -0.3649 -0.632 0.6838

Tensile Twin 3 -1 1 0 2 1 -1 0 1 -0.3419 0.5922 0.7298 0.3649 -0.632 0.6838

Tensile Twin 4 -1 0 1 2 1 0 -1 1 -0.6838 0 0.7298 0.7298 0 0.6838

Tensile Twin 5 0 -1 1 2 0 1 -1 1 -0.3419 -0.5922 0.7298 0.3649 0.632 0.6838

Tensile Twin 6 1 -1 0 2 -1 1 0 1 0.3419 -0.5922 0.7298 -0.3649 0.632 0.6838

Table 2.1: Deformation systems considered. Plane normals n and slip directions s are

expressed both in the Bravais coordinated system (a1,a2,a3, c) (sub-index brav) and in

the orthogonal system (e1, e2 and e3) (sub-index ort)

equilibrium at the end of each increment. The global solution for each time increment

is obtained iteratively using a Newton-Raphson approach. A schematic flow chart of the

calculations is depicted in Fig. 2.4 in order to clarify the different loops described below.

Each global iteration of the displacement vector corresponds, at the integration point

level, to a given deformation gradient tensor, F. Let t be the time corresponding to the

last converged increment, then, the variables Ft, Fet , Fp

t and σt (corresponding to total,

elastic and plastic deformation gradients and Cauchy stress, respectively) are known at each

integration point. In addition, the set of internal variables αt defining the CP hardening

evolution are known at time t.

At time t + ∆t, iterative predictions of the global displacement vector are obtained in

34


the context of the global Newton-Raphson scheme. A given global displacement prediction

also corresponds to a prediction of the deformation gradient at each integration point

Ft+∆t. The constitutive equation at that integration point determines the Cauchy stress,

the updated internal variables and the material tangent stiffness matrix (σt+∆t, αt+∆t and∂∆σ∂∆ε

, respectively) as function of the actual predictor Ft+∆t and the converged values of Fet

and αt.

The crystal plasticity routine solves, using an implicit scheme, an algebraical non-linear

set of equations resulting of the integration of eqs. 2.1, 2.2, 2.3, 2.9, 2.17, 2.19, 2.20 and

2.21, between t and t+ ∆t. The inputs and outputs are:

INPUTS: Ft, Fet , Ft+∆t, αt = fγit,fαt , ταc,tg

OUTPUTS: Fet+∆t, σt+∆t,

∂∆σ∂∆ε

, αt+∆t = fγαt+∆t, ...g

The starting point is the integration of the plastic velocity gradient (expression 2.3)

between t and t+ ∆t to obtain the plastic deformation gradient∫ t+∆t

t

Lpdt =

∫ t+∆t

t

FpFp−1 ! Fpt+∆t = exp(∆tLp

t+∆t)Fpt (2.22)

where the tensorial function exp() corresponds to the exponential map operator, [Souza

et al., 2008]. Combining equation 2.22 with the multiplicative decomposition, the elastic

deformation gradient at t+ ∆t can be written as

Fet+∆t = Fe

0 exp(�∆tLpt+∆t) (2.23)

where Fe0 corresponds to

Fe0 = (Ft+∆tF

−1t )Fe

t . (2.24)

If Fet+∆t is renamed simply as Fe, a tensorial residual equation can be written from

2.23 as

R(Fe) = Fe � Fe0 exp(�∆tLp(Fe)) = 0. (2.25)

and the solution of this equation provides the value of Fe. If step increments are sufficiently

small ∆tLp ! I, and the exponential function can be approached by

exp(�∆tLp) � I�∆tLp (2.26)

to reduce the computational cost. In this case, the alternative residual equation is given

by

R(Fe) = Fe � Fe0(I�∆tLp(Fe)) = 0 (2.27)

35


INPUT:, , , ,

( , , )

)

( )

) < tol

OUTPUT:,

, , ,

YES NO

=

‐ ∙R

( , , )NEW

UMAT

PRED

Figure 2.4: Flow chart of the time discretization

36


The implicit equation 2.25 is solved iteratively using a Newton-Raphson scheme, where

the new prediction of the elastic deformation gradient Fenew is given by

Fenew = Fe

old � J−1(Feold) : R(Fe

old) (2.28)

being J the corresponding Jacobian, which is a fourth-order rank tensor defined as

J =∂R(Fe)

∂Fe. (2.29)

An initial predictor for Fe is needed to solve the system. Several approaches can be

followed: a pure elastic predictor (Fe = Fe0), a pure plastic predictor (Fe = Fe

t ), and an

adaptative predictor (Fe = Fe0 exp(�∆tLp

t (Fet ))). The three approaches have been used

depending on the convergence.

In the implementation of the model, the Jacobian matrix to solve eq. 2.25 has been

analytically derived and corresponds to

Jijkl =

[∂R

∂Fe

]ijkl

= δikδjl + ∆tFe0,imEmjpq

[N∑i=1

γisi ni ∂γi

∂Fe

]pqkl

(2.30)

In the absence of hardening, the Jacobian is exact and quadratic convergence is en-

sured. When hardening is considered, the internal variables are actualized at the end of

each Newton-Raphson iteration to keep the implicit nature of eq. 2.25. In this case, the

quadratic convergence of the residual is occasionally lost.

Once the residual equation 2.25 or 2.27 has been solved, Fet+∆t and αt+∆t are obtained

and the new value of Fe is used to compute the Cauchy stress through eq. 2.18.

The final output is the material tangent matrix, defined as

C =∂∆σ

∂∆ε. (2.31)

This equation is evaluated numerically by performing six symmetric perturbations δFi,j of

the total deformation gradient at t+∆t [Kalidindi et al., 1992, Miehe et al., 1999]. The size

of the perturbation is fixed, dε, and each one corresponds to a uniaxial infinitesimal defor-

mation on the final spatial configuration. The resulting perturbed deformation gradient

can be written as

Fper:i,j = δFi,jFt+∆t. (2.32)

A new Newton-Raphson scheme is used to obtain the solution of the equation 2.25 for the

six perturbed deformation gradients, obtaining the corresponding six perturbed Cauchy

37


stress tensors σper:i,j . It should be noted that the computational cost of each of the six

evaluations of the perturbed residual is very small because the converged values Fet+∆t and

αt+∆t previously obtained can be used as predictors, reducing drastically the number of

iterations needed.

The tangent stiffness matrix is finally obtained as(∂∆σ

∂∆ε

)ijkl

�σper:k,lij � σij

dε(2.33)

2.2.2 Subroutine parameters and outputs

The model has been implemented as a material subroutine (UMAT subroutine in

ABAQUS). The only material model parameter that varies from grain to grain is the

orientation which is introduced within the abaqus input file in the definition of a material

property for each grain. All the other material data are properties of the crystal and are

given through an external file crystal.prop that is read by the subroutine only once at the

beginning of the calculation. Next, the necessary parameters in both abaqus input file and

crystal.prop file will be detailed

Definition of orientation

There are several ways of defining the orientation of a body in the three dimensional

space, forming all the possible orientations of the group SO3. In the case of polycrystalline

materials, Euler angles are normally used. These angles are necessary to rotate the global

system (X, Y, Z) to obtain the reference crystal system (X′,Y′,Z′). Each rotation is

depicted in Fig. 2.5

The rotation matrix R = Z1X2Z3, defined in Fig. 2.7 as function of the Euler angles,

relates the new orientation with global coordinate system. The columns of this matrix are

the vectors v1, v2 and v3=v1�v2 that correspond to the vectors [100], [010] and [001]

expressed in the global system (X,Y,Z), Fig. 2.6.

Definition of crystal properties

The input file (named “crystal.prop”) specifies the crystal properties. Within this file,

the following data can be found:

� the parameters of the viscous power law, γ0 and m, see Eq. 2.9

38


Figure 2.5: System reference rotation by Euler angles ϕ1, φ and ϕ2

Figure 2.6: Orientation of single crystal respect to a global system

� The total number of slip system modes N. They are four in HCP crystals: basal,

pyramidal hc+ai, prismatic and pyramidal hai.

� The number of slip systems for each slip mode.

� The number of twinning modes and its corresponding twinning systems.

39


Z1X2Z3 =

c1c3� c2s1s3 �c1s3� c2c3s1 s1s2

c3s1 + c1c2s3 c1c2c3� s1s3 �c1s2s2s3 c3s2 c2

Figure 2.7: Rotation matrix corresponding with Rotations about axes z u and z′, see

Fig. 2.5. The letters c and s stand as the cosine and sine as well as the sub-indices 1, 2

and 3 with the euler angles ϕ1, φ and ϕ2.

� The number of slip modes that can undergo re-slip.

� The plane normal and slip direction of each deformation system, (n1, n2, n3, s1, s2,

s3, slip mode number)

� The hardening coefficients qij between slip and twinning modes, for i=1,..slip+tw

modes and j=1,..slip+tw modes

� The single crystal hardening parameters for Asaro Needleman law: h0, τ0,c and τsat,

Eqs. 2.9 and 2.19.

� Internal parameters to control the Newton-Raphson iterations: toler, toler jac, niter-

max, nitermax jac, strain inc jac, implicit hard. The first two are NR tolerances for

integrating the step (from) and for integrating a perturbation of the step in order

to obtain the Jacobian matrix (from). Nitermax and nitermax jac are the maximum

number of iterations allowed for the integration of step and perturbed step, respec-

tively, and strain inc jac is the size of the strain perturbations for the jacobian.

� asl, atw parameters for Kothari and Anand phenomenological hardening laws, Eqs.

2.19, 2.20 and 2.21.

� The initial fraction of twinning for each twinning system

� The reference twinning rate f0, see eq. 2.10

� The characteristic shear of the twinning mode γtwin, see eq. 2.5

� The maximum fraction of twinning fα allowed and the value of twinning fraction

necessary to start re-slip.

An example of a crystal.prop file for an HCP crystal of a Mg alloy can be found in Appendix

A.

40

2.3 Computational homogenization framework

Subroutine Outputs

Several internal variables αt (STATEV in ABAQUS) are defined and saved at each

Gauss point for calculation and visualization purposes. A list of each of these values and

its position within the vector in which they are stored in Abaqus (statev) is given in Table

2.2:

Variable N STATEV Observations

Fet 9 1-9 -

γα 30 10-39 -

γre−slip 144 40-183 30 = Nsl +Ntw

τα 30 184-213 -

τ re−slip 144 214-357 -

Lpt 9 358-366 -

fαb 9 367-372 Twinning fraction by system

γsl−tot 1 373 Accumulated total slip plastic shear

fb sum 1 374 Twinning fraction sum by crystal

Slip phase orientation 4 375-378Euler angles and its fraction within the

crystal

Twin phase orientation (max) 4 379-382Euler angles in case of twinning system

with higher activity+fraction

Twin phase orientation (max2) 4 383-386Euler angles in case of twinning system

with second higher activity+fraction

Twin phase orientation (max3) 4 387-390Euler angles in case of twinning system

with third higher activity+fraction

γtw−tot 1 391Accumulated total twinning plastic

shear

γre−slip−tot 1 392 Accumulated total re-slip plastic shear

Slip Activities 4 393-396

Twinning Activities 1 397

Re slip Activities 4 398-401

Table 2.2: Internal variables (STATEV) saved at each point of convergence and for

each Gauss point


The mechanical behavior of different polycrystalline Mg alloys will be determined us-

ing a computational homogenization from the crystal behavior (described above) and the

41


polycrystalline microstructure. The computational homogenization techniques predict the

macroscopic behavior solving a boundary value problem on a Representative Volume Ele-

ment (RVE) of the microstructure and integrating the microfields resulting of that problem.

In this section, the finite element based computational homogenization framework devel-

oped will be presented, including the generation of the RVEs of the microstructure and

the finite element representation.

2.3.1 Microstructure representation

Two type of RVEs have been used during this work. The first type of models are ideal-

ized representations of the microstructure where each crystal in the RVE was represented

by one voxel and the models differed in the number of crystal included in the RVE, either

64, 125, 216, 512 or 1000, Figs. 2.8(a to d). Simplicity in the RVE generation and the

possibility to include a large number of grains in the RVE with limited computational cost

are the obvious advantages of this representation. However, this type of models do not

include information about the microstructural features such as grain size and shape. More-

over, it is known that the models with one voxel per crystal tends to give a stiff response

[Segurado and Llorca, 2013, Zhao et al., 2007] because the deformation is overconstrained

to maintain the compatibility between adjacent crystals. This fact is enhanced by the

poor representation of the strain fields because the linear finite elements cannot reproduce

the strain concentrations at the grain boundaries [Segurado and Llorca, 2013, Zhao et al.,

2007]. Nevertheless, this type of RVEs can provide an initial prediction of the aggregate

behavior and might result very useful do to their efficiency and automatized generation

process.

The limitations of the representation described above can be overcome with a more

realistic description of the microstructure in which each grain is discretized with several

cubic elements, Figs. 2.8(e)(f). This second type of RVE might contain an accurate

description of the polycrystalline geometry by including the actual grain shape and size

distributions when defining the element domains corresponding to each grain. Several

approaches are followed to divide the RVE in domains corresponding to different grains

[Barbe et al., 2001, Quey et al., 2011, Diard et al., 2005, Segurado and Llorca, 2013]. In

this work, the open source code Dream3D [Jackson and Groeber, 2012], was used for this

purpose. The basis of the code algorithms is to generate ellipsoids following the statistical

42


(a) (b) (c)

(d) (e) (f)

Figure 2.8: Different RVE of the polycrystal microstructure. (a)(b)(c)(d) Voxel repre-

sentation with 64, 216, 512 and 1000 cubic finite elements in which each one stands for a

grain respectively. (e)(f) Realistic RVE containing 584 and 300 crystals discretized with

� 7 and 200 cubic finite elements per grain respectively.

distribution of grains and the compact them into the RVE [Donegan et al., 2013, Tucker

et al., 2012, Wang et al., 2011].

With independence of the geometrical microstructure representation, the sole subdi-

vision of the grain with several finite elements improves the model accuracy because the

constrains imposed by neighboring grains are reduced leading to a better representation of

the strain gradients. As an example of this type of RVE’s, the models on Figs. 2.8(e)(f)

are generated using Dream3D [Jackson and Groeber, 2012] and contained 584 and 300

grains, respectively. On average, each grain was discretized with 7 (Fig. 2.8(e)) or 200

(Fig. 2.8(f)) voxels, respectively. The grains were equiaxed and the grain size followed

a log-normal distribution with an average grain volume equal to the RVE divided by the

number of grains.

43


In summary, for this second type of RVE, grain microfields are accounted and the ef-

fective behavior predictions are more accurate. However, the computational cost of these

models is much higher, specially when a large number of grains is used to represent accu-

rately the texture.

The last ingredient in the RVE generation is the representation of the actual grain

orientation distribution (texture). The method followed here was the same for both types of

RVEs. The orientation of each grain was randomly generated following a given orientation

distribution function (ODF) of the experimental textures, obtained by either EBSD or

X-ray diffraction. The ODFs definition consisted of a list of orientations (three Euler

angles) and weigths that correspond to the particular grains orientations and volumes or

to a discrete representation of the orientation space SO3 in a grid. M-tex [Bachmann

et al., 2010], was used to treat, operate and graphically represent ODFs while home-made

algorithms were used to generate the random orientations from the ODF using a Monte-

Carlo model.

Finite element simulation of the RVE

The mechanical behavior of the polycrystalline RVE was obtained by the finite element

method. The two type of voxel representations of the RVE presented above are directly

used as finite element meshes (one voxel corresponding to one cubic finite element). Pe-

riodic boundary conditions were applied on the cubic cell faces because the homogenized

polycrystal behavior derived under these conditions is always closer to the exact solution

(provided by an RVE of infinite dimensions) than those obtained with imposed displace-

ments or forces [Segurado and Llorca, 2002]. The periodic boundary conditions assume

that the RVE deforms as a jigsaw puzzle and that the whole space can be filled with a peri-

odic translation of the RVE along the three Cartesian axes. If the initial cube length is L,

and the origin of coordinates is located at one corner, the three concurrent edges of the cu-

bic RVE define an orthogonal basis e1, e2 and e3 with corresponding coordinates x1, x2, x3.

The periodic boundary conditions link the local displacement vector u of the nodes on

opposite faces of the cubic RVE with the far-field macroscopic deformation gradient F

according to,

44

2.4 Inverse optimization strategy

u(x1, x2, 0)� u(x1, x2, L) = (F� I)l3

u(x1, 0, x3)� u(x1, L, x3) = (F� I)l2

u(0, x2, x3)� u(L, x2, x3) = (F� I)l1

(2.34)

where li = Lei. The far-field deformation gradient F applied to the RVE is obtained by

prescribing the displacements of three master nodes Mi corresponding to three different

faces of the RVE,

u(Mi) = (F� I)li. (2.35)

If some components of the far-field deformation gradient are not known a priori (mixed

boundary conditions, as in under uniaxial tension), the corresponding components of the

effective stresses σ are set instead. This is carried out by applying a nodal force Pj to the

master node Mi and degree of freedom j according to

Pj(Mi) = (σei)jAi (2.36)

where Ai is the projection of the current area of the face perpendicular to ei in this

direction.

Finally, to postprocess the model results, the effective deformation gradient is obtained

by inserting the resulting displacement of master nodes on equation 2.35. The macroscopic

Cauchy stresses acting on any cube surface can be computed by dividing the reaction forces

Fj of the master nodes Mi by the actual area of the face perpendicular to that master node

Ai.

σij =FjAi

(2.37)


The mechanical behavior of the polycrystal can be obtained by the finite element sim-

ulation of the RVE and compared with the experimental results. The objective of the

optimization strategy is to obtain the set of parameters that determine the behavior of the

single crystal (the initial CRSS in each slip or twinning mode and those included in the

hardening laws given by equations 2.19, 2.20 and 2.21) which provide the best possible fit

45


between the numerical simulations of the polycrystal and the experimental data. At least

12 parameters have to be determined in the case of Mg alloys which deform by basal, pris-

matic and pyramidal hc+ ai slip together with tensile twinning, which correspond to τ0,c,

τsat and h0 for each deformation mode. The optimization process is a challenge because of

the large number of parameters and of the strong non-linearity of the problem, which is

more critical in the case of anisotropic crystals which deform by slip and twinning.

The optimization procedure is based on the Levenberg-Marquardt method [Levenberg,

1944] [Marquardt, 1963], which was adapted to be used in the context of polycrystal homog-

enization. Let xi, yi be a set of n points defining some experimental result (i.e. strain-stress

curve of the polycrystal) and let y∗i = f(xi;β) be the model prediction of that experiment

which is defined by a set of m parameters β. The Levenberg-Marquardt method is an op-

timization method to obtain the set of parameters β that minimizes the objective function

O(β) defined as

O(β) =n∑i=1

jyi � f(xi, β)j = ky � f(β)k . (2.38)

Assuming a small perturbation δ of the model parameters β, the model can be linearized

with respect to the perturbation, leading to

f(β + δ) � f(β) + Jδ (2.39)

where

Jij =∂f(xi,β)

∂βjwith 1 � i � n and 1 � j � m (2.40)

is the Jacobian matrix, obtained by evaluating the derivatives of f with respect to the set

of parameters β on the points xi. Thus, the value of the objective function O at the point

β + δ can be written as

O(β + δ) � ky � f(β)� Jδk. (2.41)

Levenberg and Marquardt minimized this objective function by adding a dumping

factor λ to the usual expression of steepest descent [Levenberg, 1944, Marquardt, 1963],

leading to the following linear set of equations

46


(JTJ + λ diag(JTJ))δ = JT [y � f(β)] (2.42)

whose solution δ provides the new set of parameters that minimizes the objective function.

In the absence of the dumping parameter λ, the iterative process to solve the non-linear set

of equations often stops because the Jacobian matrix becomes singular. This drawback can

be overcome with the addition of the (non-negative) damping factor, λ, which is adjusted

at each iteration. If the reduction of the objective function O is rapid, smaller values of

λ can be used in each iteration, bringing the algorithm closer to the fast Gauss-Newton

algorithm. On the contrary, higher λ values can be used when the objective function O is

not reduced. This procedure is repeated iteratively until the error in the objective function

reaches the desired value. In each iteration, the Jacobian matrix J (eq. 2.40) has to be

computed, the damping parameter selected and the linear set of equations in eq. 2.42

solved.

The Levenberg-Marquardt method can be easily applied if there is an analytical expres-

sion of f and its derivatives. This is not the case, however, in the case of computational

homogenization where the function y∗i = f(xi;β) is often the macroscopic stress-strain

curve of the polycrystal under certain boundary conditions which is obtained by the finite

element analysis of the RVE with a set of parameters β which define the single crystal

properties. In order to compute the Jacobian matrix (eq. 2.40), one of the parameters is

perturbed by ∆βj (in this work the value of ∆βj was of 0.05%βj, obtaining similar results

with 0.01%βj)

β∗j = β + f0, 0, � � � ,∆βj, � � � , 0gT . (2.43)

and the response of the perturbed model, f(xi;β∗j), is determined by the finite element

analysis of the RVE. This procedure is repeated for each parameter and curve in the model

and the resulting Jacobian matrix is given by,

Jij =∂f(xi;β)

∂βj� f(xi;β

∗j)� f(xi;β)

∆βj(2.44)

This step is very costly from the computational viewpoint because the number of finite

element analyses is proportional to the number of the macroscopic stress-strain curves

included in the optimization process and to the number of parameters to optimize. For

47


this reason, a hierarchical strategy was developed for the optimization strategy, that began

using very simple RVE and increased progressively the complexity of the RVE.

The ability of the Levenberg-Marquardt method to find global minimizers of the error

will depend on the parameter λ chosen on each iteration. Starting with an arbitrary value

(e.g. λ = 1) in the first iteration, three different damping parameters, corresponding to

2λ, λ and 0.5λ, are selected and the objective function (eq. 2.38) is computed for the

three damping parameters. The damping parameter which leads to a minimum error O

is selected as the starting damping parameter for the next iteration and the procedure is

repeated until the objective function O is below a given tolerance or when the difference

in the error between two consecutive steps is negligible. In the latter case, the new λ to

begin the next iteration will be either 8 times higher or smaller than the previous one. It

will be higher if the smallest error was achieved when optimization was carried out using

2λ as the damping constant and it will be smaller if the smallest error was obtained with

0.5λ.

The optimization algorithm based on the Levenberg-Marquardt method has been pro-

grammed in Python and runs as an Abaqus [Abaqus, 2013] script that reads all the nec-

essary input information (experimental stress-strain curves, microstructural information,

etc.) and automatically executes the different tasks in the optimization process: generation

of the input files for the finite element analysis with the perturbed parameters, execution

and post-processing of the finite element simulations of the RVE, assembly of the Jacobian

matrix, selection of damping parameters, solving of the linear equation set for the new

set of parameters, and checking whether the error in the objective function has reached

the desired limit to finish the process or begin another iteration, Fig. 2.9. As indicated

above, different RVEs can be used in the optimization loop to speed up the process. The

first iterations can be performed with RVEs containing 1 voxel per crystal, Figs. 2.8(a,b,c

and d) and the best parameters obtained with these RVEs are used as input for the more

realistic RVE of the microstructure, Figs. 2.8(e)(f), in the final steps of the optimization

process.

48


YES

NO → modify k

INPUT: • Experimental curves y • Model curves f(β) • Initial parameters (β)

( ) = ‖ − ( )‖

Compute Error O(β) between model and experimental curves

Calculation of perturbations: Parameters →β*

j = β + {0, 0, · · · , ∆βj , · · · , 0} Number of calculations →N = Nparameters×Ncurves For j=1 to N Abaqus calculation as function of (β*

j, input file, crystal prop, umat) Save results f(β*

j) for each j End

= ( , )

Jacobian ( ) assembly:

( + ( )) = − ( )

( ) = ‖ − ( )‖

Solve Levenberg Marquardt equation for each lambda (λ1=λ/k, λ2=λ, λ3=λ*k); (k=2, λ0=1) for i=1 to 3

( ) = + ( ) end for i=1 to 3 Abaqus calculation as function of (β'

j) Save results f(β'

i) for each i end Choose the best ∗ that minimizes ( )

( ) < ( )

NO

( ) <

END βi

’=βfinal

YES

β = β ( ) = ( ) ( ) = ( ) = ∗

New iteration:

Figure 2.9: Flow chart of Levenberg-Marquardt optimization algorithm.

49

Chapter 3Results and discussion

3.1 AZ31 Mg alloy

3.1.1 Material and processing

The methodology presented in section 2.4 of the previous chapter was applied to deter-

mine the single crystal properties of a polycrystalline AZ31 Mg alloy at room temperature

[Herrera-Solaz et al., 2014b]. The material was obtained from a plate of 25.4 mm in thick-

ness processed by hot rolling [Dogan et al., 2013]. The chemical composition is summarized

in Table 3.1.

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 0.05 max 0.05 max 0.03 max Remaining

Table 3.1: Chemical composition of the AZ31 alloy in wt.%.

The average grain size was 25 µm and the pole figure of the as-rolled material is plotted

in Fig. 3.1(a). It shows the strong basal texture typical of rolled Mg alloys, with the c

axis parallel to the normal direction (ND), as well as the spread prismatic poles along the

rolling (RD) and transverse (TD) directions.

51

Chapter 3. Results and discussion

(a)

min:0.04

max:9.3

{0001}

ND

RD

min:0.21

max:2.6

{10−10}

ND

RD

1

2

3

4

5

6

7

8

9

10

(b)

min:0.02

max:9.2

{0001}

ND

RD

min:0.19

max:2.6

{10−10}

ND

RD

1

2

3

4

5

6

7

8

9

10

Figure 3.1: Pole figures of the rolled AZ31 Mg alloy. (a) Experimental texture. (b)

Reduced equivalent initial texture with 512 orientations used as input to create the RVE.

The numbers in the legend stand for multiples of random distribution.

3.1.2 Mechanical behavior

Specimens for tension and compression experiments along different orientations were

machined from the plate. Flat dog-bone specimens were used for the tensile tests and

the dimensions of the gage section were 8 � 3 � 1.5 mm3. Compression specimens were

rectangular prisms with the dimensions of 4�4�8 mm3. Mechanical tests were carried out

using an MTS test frame at an average strain rate of 5� 10−4 s−1 . Strains were measured

with extensometers of 8 mm and 3 mm gauge length, which were attached to the tension

and compression specimens, respectively.

The material was deformed in uniaxial compression and uniaxial tension along ND and

also in uniaxial tension along RD. Additional tests were carried out in uniaxial tension in

the RD-ND plane at 45◦ from both orientations (Fig. 3.2). Three tests were carried out

in each orientation and/or loading direction (tension/compression).

The true stress - true strain curves in tension and compression in the ND, and in tension

52

3.1 AZ31 Mg alloy

Figure 3.2: Schematic of the loading directions for the mechanical tests of the rolled

plate of AZ31 Mg alloy.

in the RD directions are plotted in Fig. 3.3, together with the ones corresponding to the

tensile tests in the RD-ND plane at 45◦ from both orientations. The three experimental

curves for each test are included and they show that the experimental scatter was minimum.

These curves show the strong plastic anisotropy of Mg alloys, which is triggered by the

limited number of slip systems and by the polar nature of extension twinning, which is only

activated when deformation leads to an extension of the c axis. As a result, deformation

of wrought Mg alloys is markedly dependent on the orientation, and different slip systems

(and in different order) are activated as a function of the loading direction (either tension

or compression).

3.1.3 Optimization strategy and results

The accuracy of the optimization procedure depends on the input information used

to compute the single crystal properties, namely the direction (tension or compression)

and orientation of the mechanical tests of the polycrystal. It is obvious that if one of

the slip/twinning mode is not activated in any of the input mechanical tests, it will be

impossible the determine accurately the properties of this mode. Thus, the mechanical tests

have to be independent (should lead to the activation of different deformation mechanisms)

and the minimum number of tests to characterize the single crystal behavior will depend

on the number of active slip and twinning modes. Most papers devoted to determine

the single crystal properties of Mg alloys use two independent stress - strain curves of

the polycrystal [Fernandez et al., 2013, Wang et al., 2010, Agnew et al., 2001] but these

studies do not demostrate that this number is enough. The critical test to find out the

53


0

50

100

150

200

250

300

350

0 0.04 0.08 0.12 0.16

Tension-RDCompression-NDTension-NDTension-ND/RD 45

Stre

ss (M

Pa)

Strain

Figure 3.3: Experimental true stress - true strain curves of the AZ31 Mg alloy along

different orientations.

minimum number of independent curves is to use the single crystal parameters provided

by the optimization procedure to predict the mechanical response of the polycrystal in a

different orientation/direction.

The independence of different mechanical tests on the polycrystal can be studied by

computing the average value of the Schmid factor for each slip/twinning mode with re-

spect to the loading direction. This information can be obtained from the experimental

orientation distribution function (Fig. 3.1(a)) by averaging the Schmid factors for each

slip/twinning systems over all the grains in the microstructure for each loading case. The

average values of the maximum Schmid factor for each deformation mode are found in

Table 3.21

Tension and compression tests along ND are suitable to promote deformation by basal

and pyramidal slip but they are independent because extension twinning is likely to occur

in tension but not in compression. In addition, prismatic slip will not be dominant along

1Each deformation mode includes different slip systems. For instance, prismatic slip encompasses

(0110)[2110], (1010)[1210] and (1100)[1120]. So, the average Schmid factor corresponding to each slip

system was computed and the maximum of all the averages is listed in table 3.2 for each deformation

mode and loading case.

54

3.1 AZ31 Mg alloy

the ND direction because basal slip and twinning under tension have higher Schmid factors

but it is likely to play a major role under RD tension. Thus, these three mechanical tests

are good input candidates for the optimization procedure. The table also includes the

average values of the maximum Schmid factor for the tests carried out in the RD-ND

plane at 45◦ from both orientations. This case, in which all deformation modes can be

active, will be used to validate the optimization procedure.

Deformation mode Schmid factor

ND ND RD RD/ND45

Tension Compression Tension Tension

Basal 0.25 0.25 0.22 0.34

Prismatic 0.15 0.15 0.39 0.29

Pyramidal 0.46 0.46 0.46 0.43

Twinning 0.36 0.07 0.07 0.25

Table 3.2: Average values of the maximum Schmid factors for different deformation

modes in the polycrystalline AZ31 Mg alloy.

The effective properties of a polycrystalline AZ31 Mg alloy manufactured by hot rolling

were determined through the finite element simulation of an RVE of the microstructure.

Four different RVEs were used in a hierarchical sequence in the optimization process.

They were based on a cubic RVE discretized with cubic finite elements (voxels). In three

models, each crystal in the RVE was represented by one voxel and the models differed

in the number of crystal included in the RVE, either 64, Fig. 3.4(a), 216 or 512, Fig.

3.4(b). Simplicity in the RVE generation and the possibility to include a large number

of grains in the RVE with limited computational cost are the obvious advantages of this

representation. However, it is known that the models with one voxel per crystal tends to

give a stiff response [Segurado and Llorca, 2013, Zhao et al., 2007] because the deformation

is overconstrained to maintain the compatibility between adjacent crystals. This fact is

enhanced by the poor representation of the strain fields because the linear finite elements

cannot reproduce the strain concentrations at the grain boundaries [Segurado and Llorca,

2013, Zhao et al., 2007]. The fourth RVE model included a more realistic representation

of grain shape with � 7 voxels per grain and 512 grains to overcome these limitations, Fig.

3.4(c).

In all cases, each voxel was a cubic finite element (C3D8) in Abaqus with 8 nodes at the

55


cube corners and full integration. The orientation of each grain in the RVEs was obtained

from the experimental orientation distribution function (which describes the initial texture)

using a Monte Carlo lottery. The maximum number of orientations in the RVEs was limited

(512) and the pole figure describing the texture of an RVE with 512 grains, depicted in

Fig. 3.1(b), presented some differences with actual pole figure but captured the strong fiber

texture which was the dominant feature. In agreement with previous results [Segurado and

Llorca, 2013], it was assumed that the mechanical behavior obtained with 512 crystals was

independent of the particular random realization obtained from the ODF.

The crystal-plasticity model introduced in section 2.2 was used as the constitutive

response of the AZ31 Mg grains. The five independent elastic constants of the HCP Mg at

300 K were used here for the AZ31 Mg alloy [Zhang and Joshi, 2012]: C1111 = 59.4 GPa,

C3333 = 61.6, C1212 = 16.4 GPa , C1122 = 25.6 GPa, C1133 = 21.4 GPa. The single crystal

parameters to be obtained by the inverse optimization procedure were the initial CRSS,

τ0,c, the saturation CRSS, τsat, and initial hardening modulus, h0, for each deformation

mode considered in the model: basal, prismatic and pyramidal hc+ai slip and extension

twinning. The parameters controlling the latent-hardening, qsl−sl and qsl−tw, were 1.0 and

2.0 respectively. These values are in agreement with those used in other investigations

[Kothari and Anand, 1998, Staroselsky, 1998, Anand, 2004, Roters et al., 2010, Fernandez

et al., 2013, Wang et al., 2010] and take into account the strong hardening induced by

twinning of the traditional slip modes. The hardening exponents asl and atw were 0.6 and

1.0 respectively, which are also typical for AZ31 Mg alloy [Fernandez et al., 2013]. The rate

sensitivity exponent, m, in equation (2.9), was 0.1. With this value of m, the mechanical

response is almost independent of the strain rate when the applied strain rates in the

simulation are of the order of γ0. Reducing more the value of m impaired the convergence

and led to very similar mechanical behavior.

The finite element simulations to compute the polycrystal behavior were carried out in

Abaqus/standard [Abaqus, 2013] within the framework of the finite deformations theory

with the initial unstressed state as reference. From the available experimental results, three

tests were chosen as inputs for the optimization procedure (tension-ND, compression-ND

and tension-RD) and the fourth one (tension in the RD-ND plane at 45◦) was used to

validate the single crystal properties obtained by optimization. The objective function O

was built from the experimental stress-strain curves in three directions which, as reported

in Table 3.2, activate different deformation mechanisms. Approximately 200 points per

56

3.1 AZ31 Mg alloy

(a) (b)

(c)

Figure 3.4: Different RVE of the polycrystal microstructure for the optimization of the

AZ31 Mg alloy. (a) Voxel representation with 64 cubic finite elements in which each one

stands for a grain. (b) Voxel representation with 512 cubic finite elements, one per grain.

(c) Realistic RVE containing 512 crystals discretized with � 7 cubic finite elements per

grain.

stress-strain curve were used to build the objective function. The optimization procedure

began using the RVE with 64 grains and literature data for pure Mg (obtained from Zhang

and Joshi [2012] and shown in Table 3.3) were used as initial values for the 12 unknown

parameters that characterize the CRSSs of the three slip modes and tensile twinning.

The evolution of the error (Error O(β)) in the optimization procedure2 as function of

2The error (Error O(β)) is given by the objective function O(β), equation 2.38, divided by the number

of points n in the data set

57


the number of iterations is plotted in Fig. 3.5. It shows that the error decreased rapidly

with the number of iterations but reached a plateau after 10 iterations. At this point, the

dumping parameter λ (eq. 2.42) begins to grow and further iterations do not reduce the

error, indicating that the optimum has been reached for the RVE considered. Then, another

set of iterations was carried out using the parameters obtained in the last simulation with

the RVE containing 216 grains and afterwards with the RVE with 512 crystals. The

initial error at the beginning of each set of iteration with a new RVE was higher because

the parameters were optimized for the previous RVE but the differences were not large

indicating that the models with one voxel per grain provided a good approximation in this

strongly textured material. The final set of iterations was carried out with the realistic

RVE containing 512 grains and � 7 voxels per grain and the results obtained after two

iterations with this model were considered optimum.

0

10

20

30

40

50

0 5 10 15 20 25 30 35

64 grains / 1 voxel/grain216 grains / 1 voxel/grain512 grains / 1 voxel/grain584 grains / ≈7 voxels/grains

Erro

r O (M

Pa/p

oint

)

Number of iterations

Figure 3.5: Evolution of the objective error function per point as a function of the

number of optimization iterations for different RVEs.

The optimum values of the parameters to describe the mechanical behavior of each slip

mode and extension twinning are shown in Table 3.3 for the RVEs with 64 and 512 grains.

They were very different from the initial ones, corresponding to pure Mg, but the disparities

in the parameters obtained with different RVEs are limited, indicating that the influence

of the number of voxels per grain is limited in this particular case, in agreement with the

58

3.1 AZ31 Mg alloy

results in Fernandez et al. [2013]. This is a particular result for this material because of the

strong basal texture. Most of the grains present a similar orientation of the c axis (along

ND) and there are no important changes in stiffness between neighbor grains. Thus, the

strain microfields did not present strong discontinuities across the grain boundaries and

the stiffening associated with the poor representation of the strain gradients when the

grains are modeled with 1 finite element does not play an important role [Segurado and

Llorca, 2013, Zhao et al., 2007]. This might not be the case, however, in polycrystals with

different texture or deformation mechanisms. Nevertheless, it is important to notice that

the hierarchical procedure to start the optimization process with the simplest RVE was

very efficient in this case because it was possible to obtain a set of properties very close

to the optimum one with little computational cost. Only a few final iterations had to be

carried out with the larger RVEs to refine the results.

Parameter Deformation initial RVE

mode values 64 grains 512 grains 512 grains

64 voxels 512 voxels 4096 voxels

τ0,c

Basal 1.75 11 20 23

Prismatic 25 87 80 80

Pyramidal hc+ai 40 93 83 88

Twinning 3.5 22 34 35

τsat

Basal 40 13 23 25

Prismatic 85 101 94 94

Pyramidal hc+ai 150 168 171 179

Twinning 20 24 64 59

h0

Basal 20 1 20 20

Prismatic 1500 2831 2831 2831

Pyramidal hc+ai 3000 3817 2990 2990

Twinning 100 13 24 24

Table 3.3: Optimum values of the parameters that define the mechanical behavior of

each slip mode and extension twinning in the AZ31 Mg alloy as a function of the RVE

used in the optimization process. Magnitudes are expressed in MPa.

The results in Table 3.3 are in agreement with the general observations of the activation

of systems in randomly-oriented polycrystalline Mg alloys (including AZ31). Basal slip and

59


tension twinning are the softest deformation mechanisms at room temperature and quasi-

static strain rates, while pyramidal and prismatic slip were found to take place at much

higher stresses [Barnett et al., 2006, Hutchinson and Barnett, 2010].

The accuracy of the optimization process is clearly shown in Fig. 3.6(a), in which the

experimental and computed stress-strain curves are very close for the three orientations

in the whole deformation range. It is worth noting that this excellent agreement is only

possible because the physical mechanisms of plastic deformation and the most important

microstructural details are incorporated in the computational model. In order to validate

the optimization procedure, the tensile test in the RD-ND plane at 45◦ from both orien-

tations was simulated using the single crystal parameters obtained by optimization and

the RVE containing 512 grains. The numerical and experimental stress - strain curves are

plotted in Fig. 3.6(b).The agreement between both is very good and the average error per

point similar to the one obtained for the fitted results in the tensile ND tests (Fig. 3.6(a)).

0

50

100

150

200

250

300

0 0.02 0.04 0.06 0.08 0.1 0.12

Compression ND Tension ND Tension RD

Stre

ss (M

Pa)

Strain

(a)

0

50

100

150

200

250

300

0 0.02 0.04 0.06 0.08 0.1 0.12

Tension ND/RD 45

Stre

ss (M

Pa)

Strain

(b)

Figure 3.6: Results of the inverse optimization procedure using three stress-strain

curves (compression ND, tension ND and tension RD) as input. (a) Experimental (solid

lines) and numerical (broken lines with symbol) stress-strain curves resulting from the

optimization procedure. (b) Model prediction of the tensile test in the RD-ND plane at

45◦ from both orientations. Solid lines correspond to experimental results while broken

lines with symbols stand for the numerical simulations. The numerical results correspond

to the RVE with 512 crystals and � 7 elements per crystal.

The set of parameters obtained by the optimization procedure using three independent

stress-strain curves as input (Table 3.3) can be compared with previous data in the lit-

60

3.1 AZ31 Mg alloy

erature for rolled AZ31 Mg alloys. Our optimization strategy provided an initial CRSS

for basal slip (23 MPa) slightly lower than for extension twinning (35 MPa). Both values

are within the range of values of most of studies, that about this topic, there are in the

literature. These investigations (the majority) concluded than the CRSS for extension

twinning was approximately twice that for basal slip [Fernandez et al., 2013, Knezevic

et al., 2010, Wang et al., 2010], while others less, predicted that the CRSS for extension

twinning was significantly lower (30% to 50%) than that for basal slip [Liu et al., 2011,

Agnew et al., 2001]. Factors such as alloy composition, processing route and grain size may

account for some differences but cannot explain these large discrepancies, which should be

attributed to two factors. Firstly, the parameter identification process may not reach the

optimum solution if it is carried out manually, by a trial-and-error approach, due to the

large number of parameters and the non-linear nature of the phenomenon. Secondly, only

two experimental curves are often used to carry out the optimization [Fernandez et al.,

2013, Wang et al., 2010, Agnew et al., 2001], which correspond to tension or compression

tests along RD or ND. If only two tests are considered in the fitting procedure, it might be

possible to identify a set of parameters that provide a reasonable fit to these stress-strain

curves, but they might not be close to the optimum solution. This point will be discussed

in the following section.

In addition to the stress-strain curves, the simulations also provide information about

the activity of each deformation mode (slip and twinning) during deformation. The relative

contribution (expressed in %) of each deformation mode to the overall plastic strain is

plotted in Figs. 3.7(a), (b), (c) and (d) for the tests in tension along ND, compression

along ND, tension along RD and tension along RD-ND plane at 45◦ from both orientations

respectively. Twinning and basal slip are the dominant deformation mechanisms during

tension along ND, Fig. 3.7(a), because they present the high Schmid factors (Table 3.2)

and low CRSSs (Table 3.3). Deformation twinning is exhausted at an applied strain of

� 5%, and pyramidal and prismatic slip have to be activated to accommodate the plastic

deformation of the grains perpendicular to the c axis, because basal slip only contributes

to the plastic deformation in the basal plane. As the CRSSs of pyramidal and prismatic

slip is much higher than that of twinning (Table 3.3), the polycrystal presents a rapid

strain hardening and the overall stress-strain curve has the sigmoidal shape shown in

Fig. 3.3. On the contrary, the strong basal texture of the rolled plate limits the number

of crystals that can deform by twinning during compression along ND. Deformation is

61


initially accommodated by basal slip and twinning, but the latter is exhausted very quickly

(� 0.2%), leading to the activation of pyramidal and –to a minor extent– of prismatic slip,

Fig. 3.7(b). Pyramidal slip dominates over prismatic slip because of the texture (Table 3.2)

and the stress - strain curve presents a concave shape with continuous hardening which

is controlled by pyramidal slip, Fig. 3.3. With respect to the relative activities during

tension along the RD direction, 3.7(c), they are very similar to those reported during

compression along ND and the stress - strain curves are also very close. Finally, basal slip

and twinning are the dominant mechanisms during tension along RD-ND plane at 45◦ from

both orientations although twinning starts to be exhausted at � 5% strain. At this point,

pyramidal and prismatic slip have to be activated to accommodate the plastic deformation

of the grains perpendicular to the c axis, as it was the case under tension along ND. The

differences between tension along ND and tension along RD-ND plane at at 45◦ from both

orientations are the higher activity of basal in the latter (due to the higher Schmid factor,

Table 3.2) and the reduced twinning activity.

Another result that can be obtained from numerical simulation is the texture evolution

during deformation. The experimental and numerical pole figures of the AZ31 Mg alloy

after 10% tensile deformation along ND are plotted in Figs. 3.8 (a) and (b), respectively.

Both pole figures are in good agreement, showing the rotation basal planes from the ND

orientation to TD due to twinning, although the number of orientations in the RVE was

only 512.

The results presented in the previous paragraphs demonstrate that the inverse opti-

mization strategy based on the Levenberg-Marquardt method is able to provide a set of

parameters for the single crystal properties that can be used to predict the mechanical

behavior of the polycrystalline aggregate under different loading conditions in HCP poly-

crystals with very strong plastic anisotropy. Nevertheless, a direct comparison of the model

predictions for the CRSSs of the single crystals with experimental data is not available,

and there is always a doubt on whether this set of parameters is unique or there are other

sets than can lead to similar results. Very likely, there is not a rigorous answer to this

question (whether or not there is a unique solution to the non-linear optimization prob-

lem) and, in any case, is out of the scope of this thesis. Nevertheless, it is important to

explore the influence of two critical factors on the outcome of the optimization problem:

the input information (in terms of the stress-strain curves) and the starting points for the

optimization analysis (the initial set of parameters for the properties of the slip and twin

62

3.1 AZ31 Mg alloy

0

0.2

0.4

0.6

0.8

1

0.02 0.04 0.06 0.08

TwinningPyramidal <c+a>PrismaticBasalTwinning fraction

0

0.2

0.4

0.6

0.8

1R

elat

ive

Con

tribu

tion

Strain

f

(a)

0

0.2

0.4

0.6

0.8

1

0.02 0.04


0

0.2

0.4

0.6

0.8

1

Rel

ativ

e C

ontri

butio

n

Strain

(b)

f

0

0.2

0.4

0.6

0.8

1

0.02 0.04


0

0.2

0.4

0.6

0.8

1

Rel

ativ

e C

ontri

butio

n

Strain

(c)

f

0

0.2

0.4

0.6

0.8

1

0.02 0.04 0.06 0.08 0.1

BasalTwinningPrismaticPyramidal <c+a>Twinning fraction

0

0.2

0.4

0.6

0.8

1

Rel

ativ

e C

ontri

butio

n

Strain

(d)

f

Figure 3.7: Relative contribution of each deformation mode to the plastic strain and

volume fraction of twinned material, f , as a function of the applied strain in AZ31 Mg

alloy. (a) Tension along ND. (b) Compression along ND. (c) Tension along RD. (d)

Tension along RD-ND plane at 45◦ from both orientations

modes). They are presented in the following sections.

3.1.4 Influence of the input information

In addition to the previous optimization scenario (in which the optimization strategy

is fed with three stress - strain curves: tension along ND and RD and compression along

ND), another two scenarios were considered to study the influence of the input information

63


(a) (b)

Figure 3.8: Pole figures of the texture after 10% tensile strain along ND of the rolled

AZ31 Mg alloy. (a) Experimental results. (b) Computational homogenization results

obtained with the model with 512 orientations. The numbers in the legend stand for

multiples of random distribution.

on the final results. Only one stress-strain curve (tension along the ND) was used as input

in one of them, while two stress - strain curves (tension along ND and RD) were used in

the second one. The initial parameters in the optimization procedure for the slip and twin

modes were those corresponding to pure Mg in all cases (Table 3.3). The same hierarchical

optimization procedure was carried out in the both scenarios starting with the smallest

RVE which includes 64 grains and 1 voxel per grain, Fig. 3.6(a), and finishing with the

complex RVE including 512 grains and � 7 voxels per grain.

The influence of the input stress - strain curves (either one, two or three) on the set of

parameters that define the mechanical behavior of each slip mode and extension twinning

in the grains of the AZ31 Mg alloy are depicted in Table 3.4. The data for three curves are

copied from Table 3.3 to facilitate the comparison, which leads to the obvious conclusion

that the results obtained with one or two input stress-strain curves are different between

them and also differ from those obtained from three input stress-strain curves.

Whether the parameters obtained with one or two curves are worst than those obtained

with three curves or just different can be decided by comparing the predictions of the stress-

strain curves with the experimental ones. This information can be found in Figs. 3.9 and

3.10 for the optimizations carried out from one or two stress - strain curves, respectively.

The parameters obtained from the optimization procedure were able to provide a very good

64

3.1 AZ31 Mg alloy

Parameter Deformation Input curves

mode ND-T, RD-T, ND-C ND-T, RD-T ND-T

τ0,c

Basal 23 7 8

Prismatic 80 93 28

Pyramidal hc+ai 88 117 50

Twinning 35 31 35

τsat

Basal 25 8 106

Prismatic 94 104 82


Twinning 59 54 55

h0

Basal 20 4 35

Prismatic 2831 2824 1570


Twinning 24 306 204


each slip mode and extension twinning in the AZ31 Mg alloy as a function of the input

stress-strain curves used in the optimization procedure. Magnitudes are expressed in

MPa.

fit of the input stress-strain curve (tension along ND, Fig. 3.9(a)), but the predictions of

the three remaining stress-strain curves were very poor, particularly in the cases of tension

along RD and compression along ND, Fig. 3.9(b). This result is not surprising because the

mechanical response of the input stress-strain curve was dominated by twinning and basal

slip, Fig. 3.7(a), and it was not possible to obtain information about the CRSSs in the

pyramidal and prismatic modes. So, the CRSSs for pyramidal and prismatic slip provided

by the optimization strategy were not accurate but these plastic deformation mechanisms

are very important during tension along RD and compression along RD. Twinning was

inhibited in these cases due to the strong basal texture and the crystal deformation per-

pendicular to the basal plane has to be accommodated by either pyramidal or prismatic

slip, Figs. 3.7(b) and (c).

The predictions with the parameters obtained from the optimization strategy improved

dramatically if two stress-strain curves (tension along RD and ND) were used as input,

Fig. 3.10. Twinning and basal slip were the dominant deformation mechanisms during

65


0

50

100

150

200

250

300

0 0.02 0.04 0.06 0.08 0.1 0.12

Tension ND

Stre

ss (M

Pa)

Strain

(a)

0

50

100

150

200

250

300

0 0.02 0.04 0.06 0.08 0.1 0.12

Tension RD

Compression ND

Tension ND/RD 45

Stre

ss (M

Pa)

Strain

(b)

Figure 3.9: Results of the inverse optimization procedure using one stress-strain curves

(tension ND) as input. (a) Experimental (solid lines) and numerical (broken lines with

symbol) stress-strain curves resulting from the optimization procedure. (b) Model pre-

dictions of the compression test along ND and of the tensile test along RD and in the

RD-ND plane at 45◦ from both orientations. Solid lines correspond to experimental re-

sults while broken lines with symbols stand for the numerical simulations. The numerical

results correspond to the RVE with 512 crystals and � 7 elements per crystal.

tension along ND, while basal, pyramidal and prismatic controlled the deformation during

tension along RD, Fig. 3.7. Thus, the optimization procedure was able to obtain reliable

information about the CRSSs in the four modes and the values of the initial, τ0,c, and

saturation, τsat, CRSSs for twinning, pyramidal and prismatic slip obtained with two or

three input stress-strain curves were very similar and significant differences were only

found in the CRSSs for basal slip. The predictions of the mechanical response during

compression along ND were very good and those for tension in the RD-ND plane at 45◦

from both orientations, Fig. 3.10(b), were slightly worst than those obtained from the

parameters determined from 3 input stress-strain curves. Quantitatively, the magnitude of

the objective error function, eq. 2.38, considering all the predicted curves for each of the

cases, decreased from 31 to 25 and 11 MPa/point for the cases with one, two and three

input stress-strain curves, respectively.

These results show the critical role played by the input information to achieve accurate

results during the inverse optimization process. If the input stress-strain curve is domi-

66

3.1 AZ31 Mg alloy

0

50

100

150

200

250

300

0 0.02 0.04 0.06 0.08 0.1 0.12

Tension RDTension ND

Stre

ss (M

Pa)

Strain

(a)

0

50

100

150

200

250

300

0 0.02 0.04 0.06 0.08 0.1 0.12

Compression ND

Tension ND/RD 45

Stre

ss (M

Pa)

Strain

(b)

Figure 3.10: Results of the inverse optimization procedure using two stress-strain curves

(tension along ND and RD) as input. (a) Experimental (solid lines) and numerical

(broken lines with symbol) stress-strain curves resulting from the optimization procedure.

(b) Model predictions of the compression test along ND and of the tensile test in the RD-

ND plane at 45◦ from both orientations. Solid lines correspond to experimental results

while broken lines with symbols stand for the numerical simulations. The numerical

results correspond to the RVE with 512 crystals and � 7 elements per crystal.

nated by basal slip and twinning, the parameters obtained from the inverse optimization

strategy will never be able to predict the mechanical behavior in orientations dominated by

either prismatic or pyramidal slip. Nevertheless, two input stress-strain curves can provide

parameters very similar to those obtained with three input stress-strain curves if they are

properly chosen, i.e. one curve is controlled by basal and prismatic slip while prismatic

and pyramidal slip are dominant in the second one. It should be noticed that the input

information in the optimization process is not limited to stress-strain curves. Other data,

such as the texture evolution or the volume fraction of twinned material as a function of

the applied strain, can be included in the error function and can enhance the accuracy of

the results obtained by the inverse optimization strategy.

3.1.5 Influence of the initial set of parameters

All the inverse optimization analyses presented above for rolled AZ31 Mg alloy started

from the CRSSs of pure Mg crystals for basal, prismatic, pyramidal slip and extension

67


twinning. While this choice seems reasonable, it is important to analyze its influence

on the final results because the robustness of the optimization strategy will be seriously

compromised if the final result was strongly dependent on the starting values. To this

end, an optimization exercise was carried out with the simplest RVE which only included

64 grains and 1 voxel per grain. Three stress-strain curves were used as input (tension

along ND and RD and compression along ND) and three different set of parameters were

chosen as starting points for the optimization: one corresponding to pure Mg (Table 3.3),

another corresponding to the properties of Mg increased by a factor of 2 and a third one

in which the initial CRSS (τ0,c = 40 MPa), the saturation CRSS (τsat = 100 MPa) and the

hardening modulus (h0 = 1500 MPa) were the same for basal, prismatic and pyramidal

slip as well as twinning. The second set of parameters was chosen to increase the plastic

anisotropy of Mg while the third one was characteristic of an isotropic material.

The optimized values of the initial CRSS, τ0,c, the saturation CRSS, τsat, and the initial

hardening modulus, h0, for each deformation mode obtained from the optimization process

for the three different sets of initial parameters are plotted in Figs. 3.11(a), (b) and (c),

respectively. Despite of the initial values for the singe crystal properties, the optimization

algorithm provided similar results for the initial CRSS, τ0,c, the saturation CRSS, τsat, and

the initial hardening modulus, h0. There are only two results that are significantly different:

the saturation CRSS for twinning when the input properties are those of Mg increased by a

factor to two and the initial hardening modulus, h0 when the input properties are those of

an isotropic material. However, these differences are only apparent and do not influence the

actual response of the crystal. In the former, the initial hardening modulus is extremely

low and thus the high value of the CRSS at saturation is never reached. In the latter,

the differences between the initial and saturation CRSSs are very small and the initial

hardening modulus does not play any role.

3.2 Mg alloys containing rare earths

It has recently been shown that alloying with certain rare-earth (RE) elements might

lead to a complete elimination of the yield anisotropy at ambient temperature in wrought

Mg alloys [Ball and Prangnell, 1994, Mackenzie et al., 2007, Stanford and Barnett, 2008,

Robson et al., 2011, Hidalgo-Manrique et al., 2013]. It has been reported that RE addi-

tions influence dynamic recrystallization during processing and lead to weak deformation

68


0

50

100

150

200

Basal Pyramidal Prismatic Twinning

Mg2xMgIsotropic (t0,c=40 MPa)

t0,

c (M

Pa)

Mode

(a)

0

50

100

150

200


Mg2xMgIsotropic (tsat=100 MPa)

tsa

t (M

Pa)

Mode

(b)

0

1000

2000

3000

4000


Mg2xMgIsotropic (h0 =1500 MPa)

h0 (M

Pa)

Mode

(c)

(x1000) (x100)

Figure 3.11: Optimized values of the initial CRSS (a), the saturation CRSS (b) and the

hardening modulus after the inverse analysis, for each of the deformation modes. The

input data used in the optimization process were the ND-T, ND-C and RD-T curves.

textures during extrusion [Ball and Prangnell, 1994, Mackenzie et al., 2007, Stanford and

Barnett, 2008] and, as a result, to a more isotropic behavior during plastic deformation.

However, some studies [Robson et al., 2011, Hidalgo-Manrique et al., 2013] suggest that

such weak textures alone cannot explain the isotropy in the yield stress of Mg alloys con-

taining RE and that changes in the CRSSs of the different deformation modes with respect

to non-RE Mg alloys must also concur. These changes have not been quantified to date

69


and the origin of the influence of the RE elements on the CRSS values for Mg alloys is

currently not fully understood.

The inverse optimization methodology developed in this thesis is an ideal tool to check

this hypothesis and it was applied to determine the single crystal properties at ambient

temperature of two polycrystalline Mg alloys containing 0.5 and 1.0 wt. % Nd from the ex-

perimental stress-strain curves of uniaxial tests along different orientations with respect to

the extrusion direction (ED). The effect of Nd on the CRSSs can be quantified by compar-

ing the values obtained for these alloys with those corresponding to pure Mg [Herrera-Solaz

et al., 2014a].

3.2.1 Materials and processing

Two Mg alloys, denominated MN10 and MN11, were manufactured for this investigation

at the Magnesium Innovation Centre, Helmholtz-Zentrum Geesthacht (Germany). Their

chemical composition is summarized in Table 3.5.

Alloy Mn Nd Fe Si

MN10 1 0.5 0.15 max 0.015 max

MN11 1 1.0 0.15 max 0.015 max

Al Cu Ni Ca Mg

MN10 0.001 max 0.001 max 0.0001 max 0.02 max remaining

MN11 0.001 max 0.001 max 0.0001 max 0.02 max remaining

Table 3.5: Chemical composition of MN10 and MN11 Mg alloys (in wt.%).

The MN10 and MN11 alloys were gravity cast to produce billets for extrusion with a

diameter of 93 mm. The billets were homogenized at 350◦C during 15 h before processing.

Then, indirect extrusion of MN10 and MN11 billets was carried out at 360◦C and 350◦C,

respectively, at 2.8 mm/s, to produce round bars of 17 mm in diameter (extrusion ratio

equal to 1:30), which were subsequently air-cooled.

The as-extruded MN10 and MN11 round bars were fully recrystallized and their mi-

crostructure was formed by equiaxed grains with average diameters of 21 µm and 17 µm,

respectively. The initial texture of both alloys, measured at the center of the bars, is shown

in Fig. 3.12 by means of inverse pole figures. Both textures are weaker than those typical

of extruded non-RE containing Mg alloys [Dillamore and Roberts, 1965]. The weak texture

70


is especially pronounced in the MN11 bar while a slight tendency for the ED to be aligned

in the h1010i direction is still observed in the MN10 bar. The origin of this weak texture is

still under debate. It has been proposed that, there is a larger tendency for recrystallized

grains to nucleate at shear bands in RE-containing alloys and that the orientations of such

nuclei become more widely spread as the Nd content increases [Hidalgo-Manrique et al.,

2013]. It has also been suggested that the presence of RE solutes and intermetallic parti-

cles hinder the grain boundary mobility, thereby delaying the preferred growth of certain

orientations [Hidalgo-Manrique et al., 2013] and leading to finer grains with respect to

non-RE containing Mg alloys.

Figure 1

Figure 3.12: Microstructure in the as-extruded condition (as shown in an optical mi-

crograph) and inverse pole figure showing the orientation of the extrusion direction. (a)

MN10 Mg alloy. (b) MN11 Mg alloy


Uniaxial tension and compression tests were performed in the as-extruded bars at ambi-

ent temperature and at an initial strain rate of 10−3s−1 using an Instron universal mechan-

ical testing machine. The tensile specimens has a dog-bone shape with gauge dimensions

of 4 mm in diameter and 12 mm in length and were oriented parallel to the ED. The

compressive specimens were cylinders with 3 mm in diameter and 4.5 mm in length and

71


were machined with the loading axis in three different orientations: parallel to the ED,

inclined 45◦ with respect to the ED and perpendicular to the ED.

The four experimental stress-strain curves (tension along ED, compression along ED,

compression at 45◦ from ED and compression at 90◦ from ED) are plotted in Figs. 3.13(a)

and (b) for the MN10 and MN11 Mg alloys, respectively. Although the mechanical tests

were carried out up to very large strains (up to 20%), only the initial part of the stress-strain

curves (up to 7%) was used in the inverse optimization strategy and is plotted in Fig. 3.13.

The MN10 alloy (Fig. 3.13(a)) exhibited a yield stress asymmetry, albeit less pronounced

that that reported for non-RE Mg alloys [Yi et al., 2006]. In particular, the yield stress

in tension along the ED was 1.4 times higher than the compressive yield stress, reflecting

the combined effect of twinning polarity and texture, which results in a higher activity of

prismatic slip in tension and of tensile twinning in compression. The mechanical anisotropy

of the MN10 alloy is also evident by the different shapes of the tension and compression

stress-strain curves, the former being concave-up and the latter concave-down. On the

contrary, the MN11 alloy exhibited a very isotropic mechanical behavior (Fig. 3.13(b)).

The yield stress was very similar for all the tests and the shape of the curves is always

concave-up.

0

50

100

150

200

250

300

0 0.02 0.04 0.06 0.08

Compression 45ºEDTension EDCompression EDCompression 90ºED

Stre

ss (M

Pa)

Strain

(a) MN10

0

50

100

150

200

250

300

0 0.02 0.04 0.06 0.08

Compression 45ºEDTension EDCompression EDCompression 90ºED

Stre

ss (M

Pa)

Strain

(b) MN11

Figure 3.13: Experimental true stress-strain curves of the RE-containing Mg alloys at

ambient temperature. (a) MN10. (b) MN11.

The experimental stress-strain curves, corresponding to compression tests parallel and

perpendicular to the ED and to tension tests parallel to the ED were used as input data

72


in the optimization procedure.


The inverse optimization strategy was applied to MN10 and MN11 Mg alloys to as-

certain the influence of the Nd content on the CRSS of the different deformation modes,

namely basal, prismatic and pyramidal hc + ai slip and tensile twinning. The optimiza-

tion procedure followed a hierarchical sequence with cubic RVEs of different complexity

discretized with cubic finite elements (voxels). In the initial RVEs, each grain was rep-

resented by one voxel and the models included 125, 512 and 1000 grains per RVE. The

final optimization was carried out with an RVE created with the microstructure genera-

tor Dream3D [Jackson and Groeber, 2012] which included 300 grains and each grain was

modeled with approximately 200 voxels, Fig. 3.14. The grain shape was equiaxed, in

agreement with the experimental information, and the grain size followed a lognormal dis-

tribution characterized by µ = 1 and σ = 0.1. The orientation of each grain in the RVEs

was obtained from the experimental orientation distribution function (which describes the

initial texture) using a Monte Carlo lottery.

Figure 3.14: Cubic RVE of the microstructure including 300 crystals discretized with

200 cubic finite elements per grain

The effective behavior of MN10 and MN11 Mg alloys was determined through the CPFE

simulation of an RVE of the polycrystalline microstructure. The crystal-plasticity model

introduced in section 2 was used as the constitutive response of the MN10 and MN11

73


grains. The model accounted for the dominant deformation modes in these Mg alloys:

basal, prismatic and pyramidal hc+ai slip as well as tensile twinning. It was assumed that

the alloying elements did not modify the elastic properties and the five independent elastic

constants of the HCP Mg at 300 K were used to describe the elastic behavior[Zhang and

Joshi, 2012].

The single crystal parameters to be obtained by the inverse optimization procedure were

the initial CRSS, τ0,c, the saturation CRSS, τsat, and initial hardening modulus, h0, for

each deformation mode considered in the model: basal, prismatic and pyramidal hc + aislip and extension twinning. The initial values of these parameters in the optimization

strategy were those measured for pure Mg single crystals, that can be found in Table

3.3. The parameters controlling the latent-hardening, qsl−sl and qsl−tw, were 1.0 and 2.0

respectively, as in the case of AZ31 Mg alloy. The hardening exponents asl and atw were

0.6 and 1.0 respectively, which are also typical for AZ31 Mg alloy [Fernandez et al., 2013].

The rate sensitivity exponent, m, in equation (2.9), was 0.1. With this value of m, the

mechanical response is independent of the strain rate when the applied strain rates in the

simulation are of the order of γ0.

The finite element simulations to compute the polycrystal behavior were carried out in

Abaqus/standard [Abaqus, 2013] within the framework of the finite deformations theory

with the initial unstressed state as reference. From the available experimental results, three

mechanical tests up to an applied strain of 7% were chosen as input for the optimization

procedure (tension along ED, compression along ED and compression at 90◦ from ED)

and the fourth one (compression at 45◦ from ED) was used to validate the single crystal

properties obtained by optimization.

The input experimental stress-strain curves for the three loading cases, together with

the computed curves at the end of the optimization procedure, are plotted in Figs. 3.15(a)

and (b) for the MN10 and MN11 Mg alloys. The agreement between experimental and

numerical results is remarkable, being the average error (value of objective function divided

by the number points) always smaller than 7 MPa per point. This relatively small difference

demonstrates the ability of computational homogenization in combination with the inverse

optimization strategy to capture the mechanical response of Mg alloys. As it was previously

done for AZ31 Mg alloy and to provide further support for this statement, the fitted model

was used to predict an independent compression test performed at 45◦ from ED in both

alloys, Fig. 3.15(c). Again, the agreement between experimental and numerical results is

74


good (average error below 10 MPa per point), validating the optimization procedure and

the set of parameters obtained.

0

50

100

150

200

250

0 0.02 0.04 0.06 0.08

Compression 90º EDTension ED

Compression ED

Stre

ss (M

Pa)

Strain

(a) MN10

0

50

100

150

200

250

0 0.02 0.04 0.06 0.08

Compression 90ºED

Tension EDCompression ED

Stre

ss (M

Pa)

Strain

(b) MN11

0

50

100

150

200

250

0 0.02 0.04 0.06 0.08

MN10

MN11

Stre

ss (M

Pa)

Strain

(c)

Figure 3.15: Experimental (solid lines) and simulated (broken lines) stress-strain curves

resulting from the optimization procedure in tension along ED, compression along ED and

compression at 90◦ from ED. (a) MN10 Mg alloy. (b) MN11 Mg alloy. (c) Experimental

(solid lines) and predicted (broken lines) stress-strain curves corresponding to both alloys

tested in compression at 45◦ with respect to ED

The optimized values of the initial CRSSs, τ0,c, for the different deformation modes for

both alloys are shown in Table 3.6 and compared with those reported for pure Mg from

single crystal experiments [Zhang and Joshi, 2012]. It can be observed that Nd additions

75


CRSSs(τ0,c)(MPa) Pure Mg MN10 MN11

Basal 1.75 12 40

Tensile Twinning 3.5 24 42

Prismatic 25 65 46

Pyramidal hc+ ai 40 75 50

CRSSbasal/CRSStwinning 0.5 0.5 0.95

CRSSprism/CRSStwinning 7.1 2.7 1.1

CRSSpyr/CRSStwinning 11.4 3.1 1.2

Table 3.6: Comparison of the initial CRSSs (τ0,c) obtained by inverse optimization for

the MN10 and MN11 alloys with those measured in pure Mg single crystals [Zhang and

Joshi, 2012]

lead to an increase of the initial CRSS of all deformation modes with respect to those of

pure Mg. However, the increase in the CRSS was more pronounced for extension twinning

and basal slip. Furthermore, the CRSSbasal/CRSStwinning ratio increases notably in the RE-

containing alloys while the CRSSprism/CRSStwinning and the CRSSpyr/CRSStwinning ratios

decrease steeply. These changes are more pronounced in the MN11 alloy, in which similar

values of the initial CRSS were obtained for all active slip and twinning modes. This

is consistent with the remarkable isotropy in the mechanical behavior of this alloy. The

observed variations in the initial CRSS lead to different activities of the slip and twinning

modes, as compared with pure Mg. Firstly, the increase of the CRSSbasal/CRSStwinning

ratio is consistent with an enhancement of twinning at the expense of basal slip, which

would give rise to the concave-up shape of all the MN11 stress-strain curves, including that

corresponding to the tensile test. Promotion of twinning in the RE-containing Mg alloys

at the expense of basal slip was postulated by Hidalgo-Manrique et al. [Hidalgo-Manrique

et al., 2013] based on observations of the concave-up shape of the ambient temperature

tensile stress-strain curves of a similar MN11 alloy and it was attributed to an increase

in the CRSSbasal/CRSStwinning ratio due to the presence of intermetallic prismatic plates,

very effective for hindering basal slip [Nie, 2003]. The present numerical results confirm

the hypothesis that the CRSSbasal/CRSStwinnig increases [Hidalgo-Manrique et al., 2013]

and, furthermore, suggest that an addition of 1 wt% of Nd brings the mentioned ratio

to a value close to 1. However, since the present alloys have not been age hardened, it

is unlikely that the contribution of the precipitates in inhibiting basal slip is very high

76


and the contribution of solute atoms, which are also very effective for hardening basal slip

[Akhtar and Teghtsoonian, 1969], should be much higher.

Secondly, the reduction in the differences of the CRSS between non-basal modes and

basal slip and twinning leads to a higher activity of the former. The promotion of non-

basal slip has been largely reported for RE-containing alloys [Agnew et al., 2001, Chino

et al., 2008, Sandlobes et al., 2011, Stanford and Barnett, 2013]. However, the actual

origin of this behavior is still unclear. In HCP metals, the c/a ratio affects the difference

in the CRSS between basal and non-basal slip modes. According to some works [Agnew

et al., 2001], RE additions decrease the c/a ratio of Mg and stimulate the activation of

non-basal slip modes. It has been also put forward that RE elements influence the Peierls

potentials and the stacking fault energy on basal and non-basal planes and therefore lead

to a change in the relative CRSSs, resulting in the easier activation of non-basal slip [Chino

et al., 2008, Sandlobes et al., 2011]. The grain refinement attained in RE-containing alloys

may also lead to a higher activity of non-basal slip and not only because the differences

in CRSS between basal and non-basal slip decreases with decreasing grain size [Stanford

and Barnett, 2013], but also because the non-basal slip modes are active near the grain

boundaries, which are regions of stress concentration [Koike et al., 2003].

The experimental and simulated inverse pole figures of the compression direction after

compression along ED are shown in Fig. 3.16 for MN10 and M11 Mg alloys. Both ex-

perimental and predicted textures are very similar, evidencing further the validity of the

optimization strategy and the calculated values of CRSSs. The compression direction tends

to align mostly with the c axis in both alloys after deformation, and, less notably, with

a pyramidal direction along the h0001i-h1120i symmetry boundary of the stereographic

triangle. This is consistent with a large activity of twinning in both bars, especially in the

weakly textured MN11 alloy, which leads to an 86◦ reorientation of those grains with the

c axis inclined less than 45◦ towards the compression direction. Afterwards, grains with

basal orientations undergo a rotation towards the h1120i pole owing to pyramidal slip.

In summary, the inverse optimization strategy developed in section 2.4 of chapter 2

allowed to obtain the values of the initial CRSSs for MN10 and MN11 Mg alloys from

macroscopic testing results. The initial CRSSs were observed to change drastically with

increasing RE content. In particular, the CRSSs of basal and twinning modes as well

as the CRSSbasal/CRSStwinning ratio increased, while the CRSSprism/CRSStwinning and

CRSSpyr/CRSStwinning ratios decreased to an extent that all values become similar for

77


Figure 3

Figure 3.16: Experimental and simulated inverse pole figures showing the orientation

of the compression direction of the MN10 and MN11 Mg alloys after compression along

ED. The numbers in the legend stand for multiples of random distribution

Nd additions of 1 wt%. This is consistent with the isotropic yielding behavior observed

in the MN11 alloy. These changes in CRSSs with RE addition lead to the promotion of

twinning at the expense of basal slip and to an enhanced activity of non-basal modes.

3.3 MN11 Mg alloy at different temperatures

The results presented in the previous section have shown that the inverse optimization

strategy can be very useful to assess the influence of RE on the mechanical response

of Mg alloys at ambient temperature. Understanding the deformation mechanisms of

these materials at moderate temperatures (150◦C to 350◦) is critical, as most deformation

processing operations take place in this temperature range, but the information on the

influence of temperature on the mechanical behavior of RE-containing Mg alloys is limited

[Zhu and Nie, 2004, Bettles et al., 2009, Stanford et al., 2010, Hou et al., 2009, Azzeddine

and Bradai, 2013, Dudamell et al., 2013, Boehlert et al., 2013, Hou et al., 2012]. A key

aspect that requires clarification is the observation of an unexpected yield stress asymmetry

at high temperature, where the compressive yield stress is higher than the tensile yield

stress [Bettles et al., 2009, Hou et al., 2009]. Despite its relevance, this phenomenon has

not been extensively explored and its origin is still not understood. Thus, additional efforts

78


to fully understand the high temperature mechanical behavior of RE-containing Mg alloys

are very timely.

3.3.1 Material and processing

A MN11 Mg alloy processed at the Magnesium Innovation Centre, Helmholtz-Zentrum

Geesthacht (Germany) was used to ascertain the influence of the temperature on the me-

chanical properties. The chemical composition (see Table 3.5) and the manufacturing

process are equivalent to those described in the previous section. Billets for extrusion

produced by gravity casting were machined up to a diameter of 9 mm and homogenized

at 350◦C during 15 h before extrusion. Indirect extrusion was carried out at 275◦C at 8.8

mm/s to produce round bars of 17 mm in diameter, which corresponds to an extrusion

ratio of 1:30, which were air cooled.

A detailed microstructural characterization of the material by means of optical, scan-

ning and transmission electron microscopy was carried out in samples from the as-extruded

bar as well as from the grips of the specimens tested at different temperatures. The grain

structure perpendicular to the ED is shown in Fig. 3.17. The average grain size, measured

by the linear intercept method, has been added in an inset to each micrograph. Previous

examinations along planes parallel to ED [Hidalgo-Manrique et al., 2013] revealed that the

grains in the as-extruded bar were truly equiaxed, as befits a fully recrystallized microstruc-

ture. The average grain size remained invariant even after the material was subjected to

temperatures as high as 300◦C (Figs. 3.17(b-f)) for more than 30 minutes (i.e., the tem-

perature stabilization time plus the testing time). This indicates that the grain size of

the MN11 extrusion is thermally stable, which can be at least partially attributed to the

presence of fine and thermally stable Mn-containing particles, capable of exerting a strong

pinning force on the grain boundaries [Hidalgo-Manrique et al., 2013, 2014].

The optical micrographs of Fig. 3.17 also reveal that all the samples contain a few

coarse particles preferentially located at the grain boundaries, particularly at the highest

temperatures (see red arrows in Fig. 3.17(f)). This is consistent with the large tendency of

RE elements to segregate to the grain boundaries as a result of the radius mismatch between

RE and Mg atoms [Stanford et al., 2011]. Segregation of Nd to the grain boundaries has

been proposed to contribute to the restriction of grain growth at high temperature [Hidalgo-

Manrique et al., 2014]. Furthermore, the remarkable grey shading of the grain interiors of

79


Figure 1

Figure 2

1

Figure 3.17: Optical micrographs showing the grain structure of the MN11 Mg alloy

perpendicular to the ED. (a) As-extruded bar. (b) Grip section of the specimens tested

in tension along ED at -175◦C. (c) Idem at 50◦C, (d) Idem at 150◦C. (e) Idem at 250◦C.

(f) Idem at 300◦C. The average grain sizes are included as insets

the samples annealed at 250◦C (Fig. 3.17(e)) and 300◦C (Fig. 3.17(f)) suggests that they

also contain fine particles within the grains [Hidalgo-Manrique et al., 2014].

Analysis by transmission electron microscopy in planes perpendicular to the ED were

also performed on the as-extruded bar and on the grip sections of the specimens tested

in tension at different temperatures. Fig. 3.18 illustrates the most salient features of the

particle distributions in the samples deformed at low (-175◦C, Fig. 3.18(a)) and high

temperature (250◦C, Figs. 3.18(b-d)). Both specimens contain particles (black) with

different morphology and a maximum size of about 400 nm (Figs. 3.18(a,b)), which were

previously identified as Mn-containing particles, with a composition close to pure Mn

[Hidalgo-Manrique et al., 2014]. They are present both within the grains and at the

grain boundaries (Fig. 3.18(c)), although their spatial distribution is non-uniform. The

distribution of these particles is very similar in all the specimens, confirming their high

thermal stability. In addition, Nd-containing particles with a needle or lath morphology

and a strong orientation relation with respect to the matrix appear to nucleate close to

the Mn-containing particles and are particularly abundant in the specimens annealed at

80


250◦C (Figs. 3.18(b,d)) and 300◦C. This is consistent with the distinct grey shading that

can be observed in the corresponding optical micrographs (Figs. 3.17(e-f)). Such needles

or laths, shown in greater detail in Fig. 3.18(d), have been previously identified as Mg3Nd

phases [Hidalgo-Manrique et al., 2014], and are actually prismatic plates, the most effective

geometric obstacle for basal slip [Nie, 2003]. Finally, grain boundary precipitation of several

Nd-containing phases is also patent. Large (> 1µm) and irregularly-shaped Nd-containing

particles occasionally decorate the boundaries in all the specimens. More details about the

microstructure can be found in [Hidalgo-Manrique et al., 2015]

Figure 1

Figure 2

1

Figure 3.18: Transmission electron microscopy micrographs of the grip section of the

specimens tested in tension at 10−3s−1 and different temperatures. (a) -175◦C and (b-d)

250◦C. Micrographs are perpendicular to the ED.

The inverse pole figures in the ED of the as-extruded bar and the grip sections of the

specimens tested in tension are shown in Fig. 3.19. The as-extruded bar (Fig. 3.19(a))

exhibits a very broad angular distribution of the ED and thus displays a very weak texture

whose maximum intensity is only 2 times multiples of random distribution. Figs. 3.19(b-f)

81


also reveals that neither the immersion in nitrogen (Fig. 3.19(b)) nor the heating up to

300◦C (Figs. 3.19(c-f)) have a noticeable effect on the texture of the as-extruded bar. This

is consistent with the invariability of the average grain size at the testing temperatures

(Fig. 3.17).

Figure 3

Figure 4

2

Figure 3.19: Inverse pole figures illustrating the orientation of the ED. (a) As-extruded

bar. (b) Grip section of the specimens tested in tension along ED at -175◦C. (c) Idem at

50◦C. (d) Idem at 150◦C. (e) Idem at -250◦C. (f) Idem at -300◦C. The numbers in the

legend stand for multiples of random distribution.


Specimens for mechanical tests in tension and compression were machined from the

as-extruded bar with their loading axis parallel to the ED. The tensile specimens had a

cylindrical geometry with a gauge section of 3 mm in diameter and 10 mm in length. The

compressive specimens were also cylindrical with 3 mm in diameter and 4.5 mm in length.

Mechanical tests at an initial strain rate of 10−3s−1 were carried until failure in a Servosis

universal testing machine at -175◦C, 50◦C, 150◦C, 250◦C and 300◦C. Prior to testing,

the specimens were kept for � 20 minutes at the test temperature, which was measured

using a thermocouple clamped to the sample. The compression tests were performed

using lubrication in order to minimize friction between the sample and the anvils. For

the tests at -175◦C, the samples were tested in a liquid nitrogen bath. Otherwise, they

were tested in air within an elliptical furnace and heating was provided by four quartz

lamps. At the end of the tests, the specimens were immediately water-cooled to preserve

82


the microstructure. Additionally, several specimens were deformed in compression at 50◦C

and 250◦C at an initial strain rate of 10−3s−1 up to intermediate strains to study the

evolution of microstructure and texture with deformation.

The true stress-true strain curves corresponding to the MN11 alloy in compression and

tension along the ED are depicted in Figs. 3.20(a) and (b), respectively, as a function of

temperature. Since the alloy exhibits very weak texture, basal slip should be the dominant

deformation mechanism irrespective of the deformation mode. Therefore, concave-down

shaped curves were expected in both tension and compression. However, the shape of the

curves is concave-up in both cases up to 150◦C. Moreover, large twinning activity was found

during the first stages of deformation in compression as compared with standard Mg alloys.

This was due to the reduction in the CRSStwinning/CRSSbasal ratio with respect to that of

pure Mg [Zhang and Joshi, 2012], in agreement with the results in the previous section.

The curves are noted to have a concave-down shape above 150◦C, which is consistent with

a decline of the twinning activity.

0

50

100

150

200

250

300

350

400

0 0.1 0.2 0.3 0.4

Compression ED -175ºCCompression ED 50ºC

Compression ED 300ºCCompression ED 250ºCCompression ED 150ºC

Stre

ss (M

Pa)

Strain

(a)

0

50

100

150

200

250

300

350

400

0 0.1 0.2 0.3 0.4

Tension ED -175ºCTension ED 50ºC

Tension ED 300ºCTension ED 250ºCTension ED 150ºC

Stre

ss (M

Pa)

Strain

(b)

Figure 3.20: Experimental true stress-strain curves of the MN11 Mg alloy at different

temperatures. (a) Compression along ED. (b) Tension along ED.

The yield stresses in tension (TYS) and compression (CYS) were calculated as the true

stress at 0.2% engineering strain and are plotted in Fig. 3.21 as a function of temperature.

The TYS gradually decreased with increasing temperature while the CYS decreased from

-175◦C to 50◦C and increased thereafter up to a maximum at 250◦C before decreasing

83


again at 300◦C. It follows that the yield stress asymmetry was low at -175◦C, 50◦C and

150◦C, in agreement with earlier results on Mg-RE alloys [Ball and Prangnell, 1994, Rob-

son et al., 2011, Laser et al., 2008, Hidalgo-Manrique et al., 2013, 2014], as well as with

those in the previous section 3.2. Surprisingly, a strong reversed yield stress asymmetry

developed at higher temperatures, the CYS being significantly higher than the TYS. The

observed evolution of the yield asymmetry with temperature is very different from that

reported for wrought (rolled or extruded) conventional Mg alloys. In the latter, twinning

is easily activated under compression, but not under tension along RD or ED at ambient

temperature [Munroe et al., 1997] since the basal planes are oriented parallel to the RD or

the ED, [Barnett, 2012], resulting in a lower yield stress in compression than in tension.

The difference between the TYS and the CYS decreases as the test temperature increases

until the two become very similar [Ulacia et al., 2010, Al-Samman et al., 2010] due to the

reduction in the CRSS for the non-basal slip modes at high temperature [Barnett, 2003],

which leads to an increased activity of these modes at the expense of twinning.

40

60

80

100

120

140

-200 -100 0 100 200 300 400

Compression EDTension ED

Yie

ld S

tress

(MP

a)

Temperature (ºC)

Figure 3.21: Evolution of the yield stress in tension and compression along the ED

with temperature for the MN11 Mg alloy.

The development of a strong reversed yield stress asymmetry at moderate tempera-

tures and quasi-static rates observed in the MN11 alloy has also been reported to occur

84


in other Mg-RE alloys [Bettles et al., 2009, Hou et al., 2009]. For example, the age-

hardenable Mg-8Gd-2Y-1Nd-0.3Zn-0.6Zr (wt.%) alloy presents an isotropic behavior at

ambient temperature, but is stronger in compression than in tension at elevated tempera-

ture [Hou et al., 2009]. In the absence of twinning at such high temperatures, the reversed

yield stress asymmetry was attributed to varying interactions of dislocations with solute

atoms and precipitates in tension and compression, but no further details were given. The

age-hardenable Mg-1.7Nd-1RE-Zn-Zr (wt.%) alloy was also reported to exhibit a similar

reversed yield asymmetry, whose maximum was found to depend on the grain size [Bet-

tles et al., 2009]. Finally, the compressive yield strength became higher than tensile yield

strength at ambient temperature in the Mg-6Y-7Gd-0.5Zr (wt.%) [Robson et al., 2011],

the Mg-Y2O3 composites [Garces et al., 2006] and the Mg93Zn6Ho alloy [Singh et al., 2007]

when the grain size was reduced to about 2 µm, which led to the suppression of twin-

ning. In summary, although the origin for the reversed yield asymmetry in RE-containing

Mg alloys is still unknown, this phenomenon becomes apparent under conditions in which

twinning is difficult.

With the aim of analyzing the incidence of twinning in the deformation of MN11 Mg

alloy, the texture and microstructure of specimens deformed in compression up to an engi-

neering strain of 5% at 10−3s−1 at 50◦C and 250◦C were examined by electron backscatter

diffraction (Fig. 3.22). It can be inferred from the orientation imaging maps that twins

are indeed more abundant in the sample compressed at 50◦C (Fig. 3.23(a)) than in the

sample compressed at 250◦C (Fig. 3.23(b)). The reduced twinning activity at 250◦C tem-

perature may be consistent with a relative decrease in the CRSS of non-basal modes with

respect to that of twinning, commonly observed in Mg alloys [Barnett, 2003], and with

the hindering of twin nucleation at grain boundaries by increased Nd segregation. Grain

boundaries are generally considered to be the most common sites for twin nucleation and

thus any phenomenon that modifies the structure or chemistry of these boundaries could

also influence twin nucleation [Jain et al., 2010]. In any case twinning at 250◦C was scarce,

in agreement with previous studies [Robson et al., 2011, Bettles et al., 2009, Hou et al.,

2009, Garces et al., 2006, Singh et al., 2007], and cannot be responsible for the reversed

yield asymmetry.

More accurate information on the active deformation modes at low and high temper-

ature was obtained by means of macrotexture analysis by X-ray diffraction in the gauge

section of specimens tested at 50◦C and 250◦C. Fig. 3.23 illustrates the inverse pole figures

85


Figure 5

Figure 6

3

Figure 3.22: Electron backscatter diffraction inverse pole figure maps in the ED of spec-

imens compressed up to 5% engineering strain at 10−3s−1 at different temperatures. (a)

50◦C and (b) 250◦C. The non-indexed points are shown as black pixels. The boundaries

having a misorientation of 86◦ (� 5◦) have been depicted as white lines. The compression

axis is horizontal.

showing the orientation of the ED of specimens deformed at 50◦C in compression up to

30% engineering strain (Fig. 3.23(a)) and in tension up to failure (Fig. 3.23(b)) as well

as at 250◦C in compression up to 40% engineering strain (Fig. 3.23(c)) and in tension

up to failure (Fig. 3.23(d)). By comparison with the texture of the as-extruded material

(Fig. 3.19(a)), it can be stated that a depletion of orientations close to the h1010i-h1120iboundary takes place in compression at 50◦C (Fig. 3.23(a)), while a depletion of orienta-

tions in the vicinity of the h0001i pole takes place in tension (Fig. 3.23(b)) at the same

temperature. This is consistent with the occurrence of twinning under both tension and

compression at this temperature. Note that extension twinning, leading to a rapid 86◦

reorientation of the lattice, is only active when the resolved applied stress results in an ex-

tension of the c axis. Therefore, twinning occurs in compression in grains with their c axis

approximately perpendicular to the ED or to the compression axis (situated in the region

near the h1010i-h1120i boundary of the unit triangle) and reorients the c axis nearly paral-

lel to the compression axis [Calnan and Clews, 1951]. On the contrary, twinning occurs in

tension in crystals with their c axis approximately parallel to the ED or to the tension axis

(close to the h0001i pole of the unit triangle) and reorients the c axis nearly perpendicular

to the tension axis [Calnan and Clews, 1951]. The mentioned depletion of orientations close

to the h1010i-h1120i boundary during compression at 250◦C (Fig. 3.23(c)) is much less

86


apparent than at 50◦C (Fig. 3.23(a)), especially around the h1120i pole. This is indicative

of a lower reliance on twinning to accommodate deformation and, therefore, of a more

prominent role of non-basal slip at 250◦C. The fact that the orientation density is higher

around the h1120i pole than around the h1010i, reveals that the role of pyramidal hc+aislip is more important than that of prismatic slip. Similarly, the mentioned depletion of

orientations in the vicinity of the h0001i pole during tension at 250◦C (Fig. 3.23(d)) is

somewhat less apparent than at 50◦C (Fig. 3.23(b)). This is also indicative of a higher

reliance on pyramidal hc+ai slip at the expense of twinning to accommodate deformation.

Figure 7

Figure 8

Figure 9

4

Figure 3.23: Inverse pole figures obtained by X-ray diffraction showing the orientation

of the ED in specimens deformed under different conditions. (a) 50◦C under compression

up to 30% engineering strain. (b) 50◦C under tension up to failure. (c) 250◦C under

compression up to 40% engineering strain. (d) 250◦C under tension up to failure. The

numbers in the legend indicate multiples of random distribution.

Thus, the experimental observations seem to indicate that the inverse yield asymme-

try observed in the MN11 Mg alloy at high temperature is related to the preponderance

of pyramidal slip with respect to twinning but it is not demonstrated whether or not

this mechanism can explain the experimental results. This inverse optimization strategy

developed in this thesis was used to this end.

87



The effective behavior of the MN11 Mg alloy at different temperatures was determined

by means of the CPFE simulation of a RVE of the polycrystalline microstructure. If not

indicated otherwise, the optimization strategy is identical to that presented in the previous

section for MN10 and MN11 Mg alloys. The simulations were carried out with an RVE

including 125 grains and 1 voxel per grain. A few simulations with an RVE containing

1000 crystals were also performed and the differences with the smaller RVE were below

2%.

The experimental stress-strain curves in tension and compression (Fig. 3.20) were used

input in the optimization analysis at each temperature. The optimization procedure based

on the Levenberg-Marquardt algorithm provided sets of parameters that accurately fitted

both tension and compression curves at -175◦C, 50◦C and 150◦C. It was not possible,

however, to find sets of parameters able to fit both curves at 250◦C and 300◦C. The

reversed yield asymmetry (the compressive yield stress was higher than the tensile one) at

these temperatures could only be taken into account by the only polar mechanism in the

crystal plasticity model (tensile twinning), but twinning could hardly be responsible of this

behavior because (1) the random texture induces a very similar effect of twinning in both

tension and compression and (2) the twinning activity at high temperature was negligible

(see Fig. 3.22).

If tensile twinning cannot be the origin of the reversed yield asymmetry, the only

possibility to account for this behavior has to be found in the development of non-Schmid

effects for the activation of the pyramidal hc+ai mode. This assumption was introduced by

Bassani [Qin and Bassani, 1992] and Asaro [Dao and Asaro, 1993] in the past to account for

the anomalous deformation of some single crystals and intermetallic alloys. Accordingly,

the Schmid law was modified and the resolved shear stress in the slip system α, τα, was

substituted by an effective shear stress τ ∗α, which is given by

τ ∗α = τα + S : η (3.1)

where S is the corresponding stress tensor (see chapter 2 for more details) and

τα = S : (s n) (3.2)

is the Schmid stress, i.e. the resolved shear stress on the slip system α characterized by

88


the vector n normal to the slip plane and the vector s along the slip direction. The second

term of equation 3.1 is the non-Schmid contribution to the effective shear stress, where

η is a general non-Schmid tensor that accounts for the influence of the different stress

components on effective shear stress.

This model has been successfully applied to BCC materials [Bassani et al., 2001, Vitek

et al., 2004, Yalcinkaya et al., 2008], where the asymmetry of the dislocation cores justifies

the influence of non-Schmid stresses on the dislocation mobility, but it is not clear that

it can be applied to model the asymmetry between tension and compression in pyramidal

hc+ai systems of HCP crystals. Atomistic simulations suggested that non-Schmid stresses

might have an effect on the slip of pyramidal hc+ai dislocations connected with dynamic

dissociation [Yalcinkaya et al., 2008], while other recent studies [Yoo et al., 2001, Jones and

Hutchinson, 1981] relate a possible effect of non-Schmid stresses to the cross-slip of hc+aiscrew dislocations. In addition to the effect of non-Schmid stresses, atomistic simulations

have demonstrated that slip in the pyramidal hc+ai mode in Mg is directional [Jones and

Hutchinson, 1981]. Thus, the shear stress necessary to move a hc+ai dislocation depends

on the direction of the applied shear stress and this is probably due to the asymmetry

of the dislocation core. This phenomenon is independent of other stress components and

cannot be strictly considered a non-Schmid effect.

The directionality of the pyramidal hc+ai mode is not enough to justify the experi-

mental yield asymmetry and it is necessary to introduce non-Schmid terms to account for

the effect of hydrostatic stresses. In order to account for both effects (directionality and

hydrostatic stresses), a new definition of the effective shear stress is proposed here. Only

one additional stress term is included in the expression of the effective shear stress, which is

the projection of the global stress tensor along the c axis of the hexagonal lattice according

to:

τ ∗α = hjταj+ ηccS : (c c)isign(τα) (3.3)

where c is a unit vector along the c axis. The parameter ηcc takes into account both the

non-Schmid component in the direction of n and the directionality in the shear component.

The inverse optimization procedure was carried out using equation 3.3 to compute the

effective shear stress for the pyramidal hc+aimode in the whole temperature range. A value

of ηcc = 0.185 was selected, the minimum one to reproduce the experimental inverse yield

asymmetry of the strain-stress curves at 250◦C and 300◦C. The influence of this parameter

89


in the optimization results was negligible at lower temperatures (from -175◦C to 150◦C)

because because the activity of pyramidal hc+ai slip was minimum in this temperature

range. In fact, using ηcc = 0 or to 0.185 did not modify the results at these temperatures.

The experimental stress-strain curves in tension and compression and the curves ob-

tained from the inverse optimization procedure are plotted in Figs. 3.24(a), (b), (c), (d)

and (e) for -175◦C, 50◦C, 150◦C, 250◦C and 300◦, respectively. The agreement between

experimental and numerical results is remarkable in the whole temperature range, which

demonstrates again the ability of the optimization procedure and of the modified crystal

plasticity model to capture the anisotropic mechanical response of Mg alloys. Moreover,

inverse yield asymmetry at 250◦C and 300◦ is accurately predicted by the crystal plasticity

finite element simulations.

The values of the initial CRSS, τ0,c, the saturation CRSS, τsat, and initial hardening

modulus, h0, for each deformation mode provided by the optimization strategy are given

in Table 3.7. In addition, the evolution of the initial CRSS, τ0,c, with temperature for

each slip mode and twinning has been plotted in Fig. 3.25. It should be noted that non-

Schmid effects, as given by equation 3.3, were only considered in the simulations at 250◦C

and 300◦C because their influence was negligible at �175◦C, 50◦C and 150◦C because the

activity of pyramidal hc+ai slip was limited.

It can be observed that the CRSSs of all slip modes as well as tensile twinning were

relatively similar at 50◦C, in agreement with the results presented in the previous section

3.2. Broadly speaking, deformation from cryogenic temperature (-175◦C) up to 150◦ was

mainly controlled by basal and prismatic slip at the initial stages of deformation, while

the contribution of tensile twinning and of pyramidal hc+ai slip was limited, Figs. 3.25(a)

to (d). As a result, the yield strength in tension and compression was similar in this

temperature range, following the reported effect of RE additions on the properties of Mg

alloys. The contribution of pyramidal hc+ai became more important at the expense of

twinning in this temperature range at strains above 8% in compression, thus the strain

hardening rate in compression was much higher than in tension because the CRSS for

pyramidal hc+ai slip was much higher than that of twinning.

Nevertheless, pyramidal hc+ai slip became dominant in tension and compression at

250◦C and above, Figs. 3.26(e) and (f), from the beginning of deformation , leading to

a marked inverse yield anisotropy. The activity of pyramidal hc+ai in compression was,

however, reduced as compared with the tensile tests at the same temperature (and the

90


Parameter Deformation Temperature (◦C)

mode -175 50 150 250 300

τ0,c

Basal 33 40 40 100 100

Pyramidal hc+ai 98 52 52 33 24

Prismatic 35 36 30 70 91

Twinning 47 33 38 98 100

τsat

Basal 575 131 139 111 320

Pyramidal hc+ai 1621 80 58 142 31

Prismatic 49 40 33 91 101

Twinning 194 78 76 158 111

h0

Basal 635 408 283 248 165

Pyramidal hc+ai 10 11 38 73 124

Prismatic 464 946 1460 144 482

Twinning 771 503 200 229 355


each slip mode and extension twinning in MN11 Mg alloy as a function of temperature.

Magnitudes are expressed in MPa.

contribution of prismatic slip enhanced), because of the non-Schmid effects on the effective

shear stress on the slip plane. These results suggest that the microstructural changes that

take place within this temperature range, i.e., the precipitation of Mg3Nd plates and Nd

segregation to grain boundaries, increase significantly the CRSSs for basal and prismatic

slip as well as twinning, while CRSS for pyramidal hc+ai slip was not influenced by these

microstructural changes and decreased as the temperature increased.

91


0

100

200

300

400

500

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4


Stre

ss (M

Pa)

Strain

(a)

0

100

200

300

400

500

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4


Stre

ss (M

Pa)

Strain

(b)

0

100

200

300

400

500

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4


Stre

ss (M

Pa)

Strain

(c)

0

100

200

300

400

500

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4


Stre

ss (M

Pa)

Strain

(d)

0

100

200

300

400

500

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4


Stre

ss (M

Pa)

Strain

(e)

Figure 3.24: Experimental (solid lines) and simulated (broken lines) stress-strain curves

in tension and compression along ED in MN11 Mg alloy at different temperatures.(a)

-175◦C. (b) 50◦C. (c) 150◦C. (d) 250◦C, (e) 300◦C.

92


0

20

40

60

80

100

120

-200 -100 0 100 200 300

TwinningPrismaticPyramidal <c+a>Basal

CR

SS

(MP

a)

Temperature (ºC)

Figure 3.25: Evolution of the initial CRSS, τ0,c, with temperature for each slip mode

and twinning of MN11 Mg alloy according to the inverse optimization model.

93


0

0.2

0.4

0.6

0.8

1

0.04 0.08 0.12 0.16 0.2

TwinningPyramidal <c+a>PrismaticBasal

Rel

ativ

e C

ontri

butio

n

Strain

(a)

0

0.2

0.4

0.6

0.8

1

0.04 0.08 0.12 0.16 0.2


Rel

ativ

e C

ontri

butio

nStrain

(b)

0

0.2

0.4

0.6

0.8

1

0.05 0.1 0.15 0.2 0.25 0.3 0.35

TwinningPyramidal<c+a>PrismaticBasal

Rel

ativ

e C

ontri

butio

n

Strain

(c)

0

0.2

0.4

0.6

0.8

1

0.05 0.1 0.15 0.2 0.25 0.3 0.35


Rel

ativ

e C

ontri

butio

n

Strain

(d)

0

0.2

0.4

0.6

0.8

1

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4


Rel

ativ

e C

ontri

butio

n

Strain

(e)

0

0.2

0.4

0.6

0.8

1

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4


Rel

ativ

e C

ontri

butio

n

Strain

(f)

Figure 3.26: Relative contribution of each deformation mode to the plastic strain as

a function of temperature and loading (tension or compression along the ED) in MN11

Mg alloy. (a) -175◦C, tension. (b) -175◦C, compression. (c) -50◦C, tension. (d) 50◦C,

compression. (e) 250◦C, tension. (f) 250◦C, compression.

94

Chapter 4Conclusions and future work

4.1 Conclusions

The following conclusions can be drawn from the work presented in this thesis:

� An inverse optimization strategy has been developed to determine the single crys-

tal properties from the experimental results of the mechanical behavior of polycrystals.

The polycrystal behavior was obtained by means of the finite element simulation of an

RVE of the microstructure in which the constitutive equation of each grain was modeled

by means of crystal plasticity model which took into account the physical deformation

mechanisms, e.g. slip and twinning. The inverse problem was solved by means of the

Levenberg-Marquardt algorithm and the efficiency of the optimization strategy from the

computational viewpoint was greatly improved by means of a hierarchical approach. Thus,

the iterative optimization process began using a simple RVE in which each grain is repre-

sented by one voxel and changed to more complex and realistic RVEs once the optimization

algorithm reached the optimum solution for this RVE. Moreover, the full procedure is au-

tomatized in the form a python script to facilitate its practical implementation.

� The inverse optimization strategy was successfully applied to a rolled polycrystalline

AZ31 Mg alloy, showing a marked basal texture and a strong plastic anisotropy. Four

different deformation mechanisms (basal, prismatic and pyramidal hc+ai slip together with

tensile twinning) were included in each grain and 12 different parameters (the initial CRSS,

τ0,c, the saturation CRSS, τsat and the initial hardening modulus, h0, for each system) were

obtained from the optimization. The robustness of the inverse optimization strategy to

95

Chapter 4. Conclusions and future work

provide a good approximation of the input experimental curves was clearly shown. In

addition, the validity of the results was demonstrated by the accurate prediction of the

mechanical response of the polycrystal under loading conditions different from those used as

input for the optimization process. Finally, the critical role played by the input information

on the accuracy of the parameters obtained from the inverse optimization strategy was

proven.

� The inverse optimization strategy was also applied to study the influence of Nd

content on mechanisms controlling the mechanical behavior extruded polycrystalline Mg-

Mn (1 wt. %) alloys. It was found that the differences in the CRSSs for basal, prismatic

and pyramidal hc+ai slip as well as for tensile twinning disappeared at room temperature

when the Nd content was around 1 wt. %, leading to an isotropic mechanical response

regardless of the orientation and loading mode (tension or compression). The analysis

was extended from -175◦ C to 300◦C in the case of the Mg-Mn (1 wt. %)- Nd (1 wt.

%) alloy, which presented a marked inverse yield strength asymmetry (the compressive

yield strength being higher than the tensile one) at high temperature (� 250◦C). The

inverse optimization strategy demonstrated that the transition from an isotropic behavior

at temperatures � 150◦C to the development of a inverse yield strength asymmetry at

temperatures � 250◦C was due to a change in the dominant deformation mechanisms at

the single crystal level: plastic deformation was controlled by basal and prismatic slip and

tensile twinning below 150◦C and by pyramidal hc+ai slip at 250◦C and above. The inverse

optimization strategy demonstrated that the origin of the inverse yield stress asymmetry

at high temperature has to be primarily ascribed to non-Schmid effects on the CRSS on

pyramidal planes and thus on the mobility of the hc+ai dislocations.

4.2 Future work

The inverse optimization strategy developed in this thesis is a powerful and unique tool

to determine reliable values of the CRSSs of the different slip and twinning modes in single

crystals (which are difficult to obtain in most cases) from the mechanical response of poly-

crystals. This information is critical for mesoscale multiscale modeling strategies aimed at

providing physically-based constitutive equations for polycrystals from the microstructural

features of the material (size, shape, spatial orientation an properties of the single crystals).

The application of this strategy to different materials and loading conditions is an obvious

96

4.2 Future work

route of future work for materials science researchers and metallurgists. In addition, from

the solid mechanics perspective, there are several areas to improve the strategy developed.

They include the following topics:

� The mechanical response of each deformation mode (either slip or twinning) within

each crystal is modeled with a phenomenological law expressed by a power-law dependency

of the shear strain rate with the resolved shear stress. The implementation of physically-

based laws (including physical parameters that can be obtained form models at lower

length scales or obtained from the optimization procedure) will help to reduce the number

of parameters in the optimization strategy and to increase the range of applicability of the

results. For instance, activation energies obtained from the optimization process can be

used to predict the mechanical behavior as a function of strain rate and/or temperature

without the corresponding mechanical tests in polycrystals under these conditions.

� The current twinning model is known to predict accurately the global behavior of

polycrystals but it has many limitations from the local viewpoint. The mechanisms of

twin nucleation and growth are still under debate and this is a a very active research area.

The implementation in the crystal plasticity framework of better twining models capable of

predicting the influence of grain boundary misorientation and grain size on twin nucleation

as well as of physically-based models for twin growth and re-slip will improve the predictive

capabilities of the overall strategy.

� It has been shown that the richness of the input data is critical for the accuracy of

the parameters obtained by optimization strategy. Thus, extension of the error function

to include other experimental information (such as the evolution of the texture or of the

volume fraction of twinned material) is expected to increase greatly the accuracy.

� The implementation of spectral methods (Fast Fourier Transform) to solve the bound-

ary value problem of the RVE is important to speed up the optimization process and to

include larger RVEs which reproduce more accurately the microstructure of the polycrystal.

� Grain boundary sliding and fracture may be an important deformation mechanism

in polycrystals, particularly at high temperature and low strain rates (creep). The incor-

poration of this mechanism to the inverse optimization strategy is also very appealing to

provide information about the grain boundary properties which are also very difficult to

obtain experimentally. This will require the use of different discretization for the grains

within the RVE based on polyhedra obtained by means of Voronoi tessellation and the

introduction of the appropriate constitutive equations and/or interface elements for the

97

Chapter 4. Conclusions and future work

grain boundaries.

98

Appendix ACrystal properties

# Name: MN11_adjust

# C11, C12,C44,C13,C33,C66

58E9,25e9,16.6E9,20.8E9,61.2E9,16.6E9

#Viscoplastic law: gamma_0, rate sensitivity exponent

1.,0.10

# number of slip modes

# number of basal, pir[c+a], prys and pir[a] systems

# number of twinning modes and twinning systems

# number of re-slip systems modes

4

3,12,3,6

1,6

24

# normalv, tangent, mode ( the modes order is basal,pir[c+a],prys,pir[a],Tensile Twinning)

0,0,1,-0.866025403784439,0.5,0,1

0,0,1,-0.866025403784439,-0.5,0,1

0,0,1,0,1,0,1

0.882256902898182,0,0.470768262830565,-0.454287159974388,0.262282814100603,0.851370014570559,2

0.882256902898182,0,0.470768262830565,-0.454287159974388,-0.262282814100603,0.851370014570559,2

0.441128451449091,0.764056890574006,0.470768262830565,-0.454287159974388,-0.262282814100603,0.851370014570559,2

0.441128451449091,0.764056890574006,0.470768262830565,0,-0.524565628201207,0.851370014570559,2

-0.441128451449091,0.764056890574006,0.470768262830565,0,0.524565628201207,-0.851370014570559,2

-0.441128451449091,0.764056890574006,0.470768262830565,0.454287159974388,-0.262282814100603,0.851370014570559,2

0.882256902898182,0,-0.470768262830565,0.454287159974388,0.262282814100603,0.851370014570559,2

0.882256902898182,0,-0.470768262830565,0.454287159974388,-0.262282814100603,0.851370014570559,2

0.441128451449091,0.764056890574006,-0.470768262830565,0,0.524565628201207,0.851370014570559,2

0.441128451449091,0.764056890574006,-0.470768262830565,0.454287159974388,0.262282814100603,0.851370014570559,2

-0.441128451449091,0.764056890574006,-0.470768262830565,0,0.524565628201207,0.851370014570559,2

-0.441128451449091,0.764056890574006,-0.470768262830565,-0.454287159974388,0.262282814100603,0.851370014570559,2

1,0,0,0,1,0,3

0.5,0.866025403784439,0,-0.866025403784439,0.5,0,3

99

Appendix A. Crystal properties

0.5,-0.866025403784439,0,-0.866025403784439,-0.5,0,3

0.882256902898182,0,0.470768262830565,0,1,0,4

0.441128451449091,0.764056890574006,0.470768262830565,-0.866025403784439,0.5,0,4

-0.441128451449091,0.764056890574006,0.470768262830565,-0.866025403784439,-0.5,0,4

0.882256902898182,0,-0.470768262830565,0,1,0,4

0.441128451449091,0.764056890574006,-0.470768262830565,-0.866025403784439,0.5,0,4

-0.441128451449091,0.764056890574006,-0.470768262830565,-0.866025403784439,-0.5,0,4

0.683762260317027,0,0.729704852228729,-0.729704852228729,0,0.683762260317027,5

0.341881130158513,0.592155487583613,0.729704852228729,-0.364852426114364,-0.631942939294849,0.683762260317027,5

-0.341881130158513,0.592155487583613,0.729704852228729,0.364852426114364,-0.631942939294849,0.683762260317027,5

-0.683762260317027,0,0.729704852228729,0.729704852228729,0,0.683762260317027,5

-0.341881130158513,-0.592155487583613,0.729704852228729,0.364852426114364,0.631942939294849,0.683762260317027,5

0.341881130158513,-0.592155487583613,0.729704852228729,-0.364852426114364,0.631942939294849,0.683762260317027,5

# Matrix hardening coefficients qi,j,k,l,m,n (From qbasal-basal to qtw-tw)

1.0000E+00,1.0000E+00,1.0000E+00,1.0000E+00,2.0000E+00

1.0000E+00,1.0000E+00,1.0000E+00,1.0000E+00,2.0000E+00

1.0000E+00,1.0000E+00,1.0000E+00,1.0000E+00,2.0000E+00

1.0000E+00,1.0000E+00,1.0000E+00,1.0000E+00,2.0000E+00

0,0,0,0,1.0000E+00

# Single crystal behavior based on Asaro-Needleman: tau0,taus,h0

100000000,319610994,1780000000,0

12721486.2,28620094.6,50000000,0

40000000,77061942,1500000000,0

15000000000,20000000000,1500000000,0

100000000,111000000,354807542,0

# Control of subroutine: TOLER, TOLER_JAC, Nmax iter, Nmax iter JAC,strain_inc, IMPLICIT HARD (yes=1)

1d-7,1D-7,250,5,1D-6,0

# Single crystal behavior based on Kothari: a_sl, a_tw

6.0000E-01,1.0000E+00

# Definition of twinning

# fb(beta-i) (initial fraction of twinning for each twinning system)

0.0

0.0

0.0

0.0

0.0

0.0

# fdotb(CT),fdotb(TT) (fdotbA·gammatwin=gamma_0=1.0)7.752

# gammatwin

0.129

# fbsum_max, fb_acti_re_slip

0.80,0.80

100

Bibliography

Abaqus (2013). Analysis User’s manual. Abaqus version 6.13.

Agnew, S., Yoo, M., and Tome, C. (2001). Application of texture simulation to under-

standing mechanical behavior of Mg and solid solution alloys containing Li or Y. Acta

Materialia, 49(20):4277 – 4289.

Akhtar, A. and Teghtsoonian, E. (1969). Solid solution strengthening of magnesium single

crystals-I alloying behaviour in basal slip. Acta Metallurgica, 17(11):1339 – 1349.

Al-Samman, T., Li, X., and Chowdhury, S. G. (2010). Orientation dependent slip and

twinning during compression and tension of strongly textured magnesium AZ31 alloy.

Materials Science and Engineering: A, 527(15):3450 – 3463.

Alam, M. E., Han, S., Nguyen, Q. B., Hamouda, A. M. S., and Gupta, M. (2011). De-

velopment of new magnesium based alloys and their nanocomposites. Journal of Alloys

and Compounds, 509(34):8522 – 8529.

Anand, L. (2004). Single-crystal elasto-viscoplasticity: Application to texture evolution

in polycrystalline metals at large strains. Computer Methods in Applied Mechanics and

Engineering, 193(48-51):5359 – 5383. Advances in Computational Plasticity.

Arsenlis, A. and Parks, D. M. (2002). Modeling the evolution of crystallographic dislocation

density in crystal plasticity. Journal of the Mechanics and Physics of Solids, 50(9):1979

– 2009.

Asaro, R. and Needleman, A. (1985). Overview no. 42 Texture development and strain

hardening in rate dependent polycrystals. Acta Metallurgica, 33(6):923 – 953.

Asaro, R. and Rice, J. (1977). Strain localization in ductile single crystals. Journal of the

Mechanics and Physics of Solids, 25(5):309 – 338.

101

Bibliography

Avedesian, M. M. and Baker, H. (1999). Magnesium and Magnesium Alloys. ASM Inter-

national.

Azzeddine, H. and Bradai, D. (2013). On some aspects of compressive properties and

serrated flow in Mg-Y-Nd-Zr alloy. Journal of Rare Earths, 31(8):804 – 810.

Bachmann, F., Hielscher, R., and Schaeben, H. (2010). Texture analysis with MTEX-Free

and Open Source Software Toolbox. Solid State Phenomena, 160:63–68.

Ball, E. and Prangnell, P. (1994). Tensile-compressive yield asymmetries in high strength

wrought magnesium alloys. Scripta Metallurgica et Materialia, 31(2):111 – 116.

Barbe, F., Decker, L., Jeulin, D., and Cailletaud, G. (2001). Intergranular and intragranular

behavior of polycrystalline aggregates. Part 1: F.E. model. International Journal of

Plasticity, 17(4):513 – 536.

Barnett, M. (2003). A taylor model based description of the proof stress of magnesium

AZ31 during hot working. Metallurgical and Materials Transactions A: Physical Metal-

lurgy and Materials Science, 34(9):1799–1806.

Barnett, M. (2012). Twinning and its role in wrought magnesium alloys. Advances in

Wrought Magnesium Alloys, pages 105 – 143.

Barnett, M., Keshavarz, Z., and Ma, X. (2006). A semianalytical sachs model for the flow

stress of a magnesium alloy. Metallurgical and Materials Transactions A, 37A(4):2283 –

2293.

Barton, N. R., Arsenlis, A., and Marian, J. (2013). A polycrystal plasticity model of

strain localization in irradiated iron. Journal of the Mechanics and Physics of Solids,

61:341–351.

Bassani, J., Ito, K., and Vitek, V. (2001). Complex macroscopic plastic flow arising from

non-planar dislocation core structures. Materials Science and Engineering: A, 319-321:97

– 101.

Bassani, J. L. and Wu, T.-Y. (1991). Latent hardening in single crystals II. Analytical

characterization and predictions. Proceedings of the Royal Society of London. Series A:

Mathematical and Physical Sciences, 435(1893):21–41.

102

Bibliography

Bettles, C., Gibson, M., and Zhu, S. (2009). Microstructure and mechanical behaviour of

an elevated temperature Mg-rare earth based alloy. Materials Science and Engineering:

A, 505(1-2):6 – 12.

Bishop, J. and Hill, R. (1951). XLVI. A theory of the plastic distortion of a polycrystalline

aggregate under combined stresses. Philosophical Magazine Series 7, 42(327):414–427.

Boehlert, C., Chen, Z., Chakkedath, A., Gutierrez-Urrutia, I., Llorca, J., Bohlen, J., Yi,

S., Letzig, D., and Perez-Prado, M. (2013). In situ analysis of the tensile deformation

mechanisms in extruded Mg-1Mn-1Nd (wt%). Philosophical Magazine, 93(6):598–617.

Bravais, A. (1850). Memoire sur les systemes formes par les points distribues regulierement

sur un plan ou dans l’espace. J. Ecole Polytech, 19:1–128.

Calnan, E. and Clews, C. (1951). XCIII. The development of deformation textures in

metals. part III. Hexagonal structures. Philosophical Magazine Series 7, 42(331):919–

931.

Capolungo, L., Beyerlein, I., and Tome, C. (2009). Slip-assisted twin growth in hexagonal

close-packed metals. Scripta Materialia, 60(1):32 – 35.

Cheong, K.-S. and Busso, E. P. (2004). Discrete dislocation density modelling of single

phase FCC polycrystal aggregates. Acta Materialia, 52(19):5665 – 5675.

Chino, Y., Kado, M., and Mabuchi, M. (2008). Enhancement of tensile ductility and stretch

formability of magnesium by addition of 0.2 wt%(0.035 at%)Ce. Materials Science and

Engineering: A, 494(1-2):343 – 349.

Dao, M. and Asaro, R. (1993). Non-schmid effects and localized plastic flow in intermetallic

alloys. Materials Science and Engineering: A, 170(1-2):143 – 160.

Dennis, J. and Schnabel, R. (1996). Numerical Methods for Unconstrained Optimization

and Nonlinear Equations. Classics in Applied Mathematics. Society for Industrial and

Applied Mathematics.

Diard, O., Leclercq, S., Rousselier, G., and Cailletaud, G. (2005). Evaluation of finite

element based analysis of 3D multicrystalline aggregates plasticity: Application to crys-

tal plasticity model identification and the study of stress and strain fields near grain

boundaries. International Journal of Plasticity, 21(4):691 – 722.

103

Bibliography

Dillamore, I. and Roberts, W. (1965). Preferred orientation in wrought and anneal-metals.

Acta Metallurgica, 10:271 – 280.

Dogan, E., I.Karaman, Foley, D., and Hartwig, K. (2013). Texture modification and mi-

crostructural design of AZ31 Mg alloy plate for better formability. Magnesium Workshop

Madrid.

Donegan, S., Tucker, J., Rollett, A., Barmak, K., and Groeber, M. (2013). Extreme value

analysis of tail departure from log-normality in experimental and simulated grain size

distributions. Acta Materialia, 61(15):5595 – 5604.

Dudamell, N., Hidalgo-Manrique, P., Chakkedath, A., Chen, Z., Boehlert, C., Galvez, F.,

Yi, S., Bohlen, J., Letzig, D., and Perez-Prado, M. (2013). Influence of strain rate on the

twin and slip activity of a magnesium alloy containing neodymium. Materials Science

and Engineering: A, 583:220 – 231.

Earl D. Mcbride, M. (1938). Absorbable metal in bone surgery: A further report on the use

of magnesium alloys. Journal of the American Medical Association, 111(27):2464–2467.

Eidel, B. (2011). Crystal plasticity finite-element analysis versus experimental results of

pyramidal indentation into (0 0 1) fcc single crystal. Acta Materialia, 59(4):1761 – 1771.

Eshelby, J. (1957). The determination of the elastic field of an ellipsoidal inclusion and

related problems. Procroya, 252:561–569.

Fernandez, A., Jerusalem, A., Gutierrez-Urrutia, I., and Perez-Prado, M. (2013). Three-

dimensional investigation of grain boundary-twin interactions in a Mg AZ31 alloy by

electron backscatter diffraction and continuum modeling. Acta Materialia, 61(20):7679

– 7692.

Fernandez, A., Perez-Prado, M., Wei, Y., and Jerusalem, A. (2011). Continuum modeling

of the response of a Mg alloy AZ31 rolled sheet during uniaxial deformation. Interna-

tional Journal of Plasticity, 27(11):1739 – 1757.

Garces, G., Rodrıguez, M., Perez, P., and Adeva, P. (2006). Effect of volume fraction

and particle size on the microstructure and plastic deformation of Mg-Y2o3 composites.

Materials Science and Engineering: A, 419(1-2):357 – 364.

104

Bibliography

Gianola, D. and Eberl, C. (2009). Micro and nanoscale tensile testing of materials. JOM,

61(3):24–35.

Gupta, M., Sharon, and N.M.L. (2011). Magnesium, Magnesium Alloys, and Magnesium

Composites. Wiley.

Hazanov, S. and Huet, C. (1994). Order relationships for boundary conditions effect in

heterogeneous bodies smaller than the representative volume. Journal of the Mechanics

and Physics of Solids, 42(12):1995 – 2011.

Herrera-Solaz, V., Hidalgo-Manrique, P., Perez-Prado, M., Letzig, D., Llorca, J., and

Segurado, J. (2014a). Effect of rare earth additions on the critical resolved shear stresses

of magnesium alloys. Materials Letters, 128:199 – 203.

Herrera-Solaz, V., LLorca, J., Dogan, E., Karaman, I., and Segurado, J. (2014b). An

inverse optimization strategy to determine single crystal mechanical behavior from poly-

crystal tests: Application to AZ31 Mg alloy. International Journal of Plasticity, 57:1 –

15.

Hidalgo-Manrique, P., Herrera-Solaz, V., Segurado, J., Llorca, J., Galvez, F., Ruano, O.,

Yid, S., and Perez-Prado, M. (2015). Origin of the reversed yield asymmetry in Mg-rare

earth alloys at high temperature. Acta Materialia. Submitted for publication.

Hidalgo-Manrique, P., Yi, S., Bohlen, J., Letzig, D., and Perez-Prado, M. (2013). Effect of

Nd additions on extrusion texture development and on slip activity in a Mg�Mn alloy.

Metallurgical and Materials Transactions A, 44(10):4819–4829.

Hidalgo-Manrique, P., Yi, S., Bohlen, J., Letzig, D., and Perez-Prado, M. (2014). Con-

trol of the Mechanical Asymmetry in an Extruded MN11 Alloy by Static Annealing.


Hill, R. (1963). Elastic properties of reinforced solids: Some theoretical principles. Journal

of the Mechanics and Physics of Solids, 11(5):357 – 372.

Hill, R. (1966). Generalized constitutive relations for incremental deformation of metal

crystals by multislip. Journal of the Mechanics and Physics of Solids, 14(2):95 – 102.

105

Bibliography

Hill, R. and Rice, J. (1972). Constitutive analysis of elastic-plastic crystals at arbitrary

strain. Journal of the Mechanics and Physics of Solids, 20(6):401 – 413.

Hou, X., Cao, Z., Sun, X., Wang, L., and Wang, L. (2012). Twinning and dynamic

precipitation upon hot compression of a Mg-Gd-Y-Nd-Zr alloy. Journal of Alloys and

Compounds, 525:103 – 109.

Hou, X., Peng, Q., Cao, Z., Xu, S., Kamado, S., Wang, L., Wu, Y., and Wang, L. (2009).

Structure and mechanical properties of extruded Mg-gd based alloy sheet. Materials

Science and Engineering: A, 520(1-2):162 – 167.

Huet, C. (1990). Application of variational concepts to size effects in elastic heterogeneous

bodies. Journal of the Mechanics and Physics of Solids, 38(6):813 – 841.

Hutchinson, J. (1976). Bounds and self-consistent estimates for creep of polycrystalline

materials. Proceedings of the Royal Society of London A, 348:101 – 127.

Hutchinson, W. and Barnett, M. (2010). Effective values of critical resolved shear stress for

slip in polycrystalline magnesium and other hcp metals. Scripta Materialia, 63(7):737 –

740. Viewpoint set no. 47 Magnesium Alloy Science and Technology.

Jackson, M. and Groeber, M. (2012). Dream.3d. http://dream3d.bluequartz.net.

Jain, J., Poole, W., Sinclair, C., and Gharghouri, M. (2010). Reducing the tension-

compression yield asymmetry in a Mg-8Al-0.5Zn alloy via precipitation. Scripta Mate-

rialia, 62(5):301 – 304.

Jiang, J., Godfrey, A., Liu, W., and Liu, Q. (2008). Microtexture evolution via deformation

twinning and slip during compression of magnesium alloy AZ31. Materials Science and

Engineering: A, 483a484:576 – 579. 14th International Conference on the Strength of

Materials.

Jones, I. and Hutchinson, W. (1981). Stress-state dependence of slip in Titanium-6Al-4V

and other H.C.P. metals. Acta Metallurgica, 29(6):951 – 968.

Kacher, J. and Minor, A. M. (2014). Twin boundary interactions with grain boundaries

investigated in pure rhenium. Acta Materialia, 81:1 – 8.

106

Bibliography

Kalidindi, S., Bronkhorst, C., and Anand, L. (1992). Crystallographic texture evolution in

bulk deformation processing of FCC metals. Journal of the Mechanics and Physics of

Solids, 40(3):537 – 569.

Kalidindi, S. R. (1998). Incorporation of deformation twinning in crystal plasticity models.

Journal of the Mechanics and Physics of Solids, 46(2):267 – 290.

Kim, G. S. (2011). Etudes submicroniques de la plasticite du monocristal de Mg. Theses,

Univerite De Grenoble.

Kim, J. J. and Han, D. S. (2008). Recent development and applications of magnesium alloys

in the Hyundai and Kia Motors Corporation. Materials Transactions, 49(5):894–897.

Knezevic, M., Levinson, A., Harris, R., Mishra, R. K., Doherty, R. D., and Kalidindi,

S. R. (2010). Deformation twinning in AZ31 Influence on strain hardening and texture

evolution. Acta Materialia, 58(19):6230 – 6242.

Koike, J., Kobayashi, T., Mukai, T., Watanabe, H., Suzuki, M., Maruyama, K., and

Higashi, K. (2003). The activity of non-basal slip systems and dynamic recovery at

room temperature in fine-grained AZ31B magnesium alloys. Acta Materialia, 51(7):2055

– 2065.

Kothari, M. and Anand, L. (1998). Elasto-viscoplastic constitutive equations for polycrys-

talline metals: Application to tantalum. Journal of the Mechanics and Physics of Solids,

46(1):51 – 83.

Kroner, E. (1961). Zur plastischen verformung des vielkristalls. Acta Metallurgica, 9(2):155

– 161.

Laser, T., Hartig, C., Nurnberg, M., Letzig, D., and Bormann, R. (2008). The influence of

calcium and cerium mischmetal on the microstructural evolution of Mg-3Al-1Zn during

extrusion and resulting mechanical properties. Acta Materialia, 56(12):2791 – 2798.

Lebensohn, R., Rollett, A., and Suquet, P. (2011). Fast fourier transform-based modeling

for the determination of micromechanical fields in polycrystals. Jom, 63(3):13–18.

Lebensohn, R. and Tome, C. (1993). A self-consistent anisotropic approach for the sim-

ulation of plastic deformation and texture development of polycrystals: Application to

zirconium alloys. Acta Metallurgica et Materialia, 41(9):2611 – 2624.

107

Bibliography

Levenberg, K. (1944). A method for the solution of certain non-linear problems in least

squares. Quarterly of Applied Mathematics, 2:164 – 168.

Leyson, G., Curtin, W., Hector, L., and Woodward, C. (2010). Quantitative prediction of

solute strengthening in aluminium alloys. Nature Materials, 9(9):750–755.

Liu, P., chang Xin, Y., and Liu, Q. (2011). Plastic anisotropy and fracture behavior of

AZ31 magnesium alloy. Transactions of Nonferrous Metals Society of China, 21(4):880

– 884.

Liu, Y., Wang, B., Yoshino, M., Roy, S., Lu, H., and Komanduri, R. (2005). Combined nu-

merical simulation and nanoindentation for determining mechanical properties of single

crystal copper at mesoscale. Journal of the Mechanics and Physics of Solids, 53(12):2718

– 2741.

Ludwig, W., King, A., Reischig, P., Herbig, M., Lauridsen, E. M., Schmidt, S., Proudhon,

H., Forest, S., Cloetens, P., Rolland du Roscoat, S., Buffiere, J. Y., Marrow, T. J., and

Poulsen, H. F. (2009). New opportunities for 3D materials science of polycrystalline

materials at the micrometre lengthscale by combined use of X-ray diffraction and X-ray

imaging. Materials Science and Engineering, A524:69–76.

Lyon, P., Wilks, T., and Syed, I. (2005). The influence of alloying elements and heat

treatment upon properties of Elektron 21 (EV31A) alloy. TMS Magnesium Technology,

pages 303 – 308.

Ma, A., Roters, F., and Raabe, D. (2006). A dislocation density based constitutive model

for crystal plasticity FEM including geometrically necessary dislocations. Acta Materi-

alia, 54(8):2169 – 2179.

Mackenzie, L. W. F., Davis, B., Humphreys, F. J., and Lorimer, G. W. (2007). The

deformation, recrystallisation and texture of three magnesium alloy extrusions. Materials

Science and Technology, 23(10):1173 – 1180.

Mandel, J. (1972). Plasticite classique et viscoplasticite: course held at the Department

of Mechanics of Solids, September-October, 1971. Courses and lectures - International

Centre for Mechanical Sciences. Springer-Verlag.

108

Bibliography

Marquardt, D. (1963). An algorithm for least-squares estimation of nonlinear parameters.

SIAM Journal on Applied Mathematics, 11:431 – 441.

Michel, J., Moulinec, H., and Suquet, P. (1999). Effective properties of composite materials

with periodic microstructure: a computetional approach. Compuational Methods in

Applied Mechanics and Engineering, 172:109–143.

Miehe, C., Schotte, J., and Lambrecht, M. (2002). Homogenization of inelastic solid ma-

terials at finite strains based on incremental minimization principles. Application to

the texture analysis of polycrystals. Journal of the Mechanics and Physics of Solids,

50(10):2123 – 2167.

Miehe, C., Schroder, J., and Schotte, J. (1999). Computational homogenization analysis in

finite plasticity simulation of texture development in polycrystalline materials. Computer

Methods in Applied Mechanics and Engineering, 171(3-4):387 – 418.

Molinari, A., Canova, G., and Ahzi, S. (1987). A self consistent approach of the large

deformation polycrystal viscoplasticity. Acta Metallurgica, 35(12):2983 – 2994.

Munroe, N., Tan, X., and Gu, H. (1997). Orientation dependence of slip and twinning in

HCP metals. Scripta Materialia, 36(12):1383 – 1386.

Nie, J. (2003). Effects of precipitate shape and orientation on dispersion strengthening in

magnesium alloys. Scripta Materialia, 48(8):1009 – 1015. ViewPoint Set No. 29 Phase

Transformations and Deformations in Magnesium Alloys.

Okamoto, H. (1998). Phase diagrams of Binary Magnesium alloys. ASM International.

Pekguleryuz, M., Kainer, K., and Kaya, A. (2013). Fundamentals of Magnesium Alloy

Metallurgy. Woodhead Publishing Series in Metals and Surface Engineering. Elsevier

Science.

Prasad, K. E., Rajesh, K., and Ramamurty, U. (2014). Micropillar and macropillar com-

pression responses of magnesium single crystals oriented for single slip or extension

twinning. Acta Materialia, 65:316 – 325.

Qin, Q. and Bassani, J. L. (1992). Non-associated plastic flow in single crystals. Journal

of the Mechanics and Physics of Solids, 40(4):835 – 862.

109

Bibliography

Quey, R., Dawson, P., and Barbe, F. (2011). Large-scale 3D random polycrystals for

the finite element method: Generation, meshing and remeshing. Computer Methods in

Applied Mechanics and Engineering, 200(17-20):1729 – 1745.

Reed-Hill, R. and Robertson, W. (1957a). Additional modes of deformation twinning in

magnesium. Acta Metallurgica, 5(12):717 – 727.

Reed-Hill, R. and Robertson, W. (1957b). The crystallographic characteristics of fracture

in magnesium single crystals. Acta Metallurgica, 5(12):728 – 737.

Remy, L. (1981). The interaction between slip and twinning systems and the influence of

twinning on the mechanical behavior of fcc metals and alloys. Metallurgical Transactions

A, 12(3):387–408.

Rice, J. (1971). Inelastic constitutive relations for solids: An internal-variable theory and

its application to metal plasticity. Journal of the Mechanics and Physics of Solids,

19(6):433 – 455.

Robertson, I. M., Schuh, C. A., Vetrano, J. S., Browning, N. D., Field, D. P., Jensen,

D. J., Miller, M. K., Baker, I., Dunand, D. C., Dunin-Borkowski, R., Kabius, B., Lozano-

Perez, T. K. T. S., Misra, A., Rohrer, G. S., Rollett, A. D., Taheri, M. L., Thompson,

G. B., Uchic, M., Wang, X. L., and Was, G. (2011). Towards an integrated materials

characterization toolbox. Journal of Materials Research, 26:1341–1383.

Robson, J., Twier, A., Lorimer, G., and Rogers, P. (2011). Effect of extrusion conditions

on microstructure, texture, and yield asymmetry in Mg�6Y� 7Gd� 0.5 wt%Zr alloy.

Materials Science and Engineering: A, 528(24):7247 – 7256.

Roters, F., Eisenlohr, P., Hantcherli, L., Tjahjanto, D., Bieler, T., and Raabe, D. (2010).

Overview of constitutive laws, kinematics, homogenization and multiscale methods in

crystal plasticity finite-element modeling: Theory, experiments, applications. Acta Ma-

terialia, 58(4):1152 – 1211.

Sachs, G. (1928). Plasticity problems in metals. Trans. Faraday Soc., 24:84–92.

Salem, A., Kalidindi, S., and Semiatin, S. (2005). Strain hardening due to deformation

twinning in α-titanium: Constitutive relations and crystal-plasticity modeling. Acta

Materialia, 53(12):3495 – 3502.

110

Bibliography

Sanchez-Martın, R., Perez-Prado, M., Segurado, J., Bohlen, J., Gutierrez-Urrutia, I.,

Llorca, J., and Molina-Aldareguia, J. (2014). Measuring the critical resolved shear

stresses in Mg alloys by instrumented nanoindentation. Acta Materialia, 71:283 – 292.

Sandlobes, S., Zaefferer, S., Schestakow, I., Yi, S., and Gonzalez-Martinez, R. (2011). On

the role of non-basal deformation mechanisms for the ductility of Mg and Mg-Y alloys.

Acta Materialia, 59(2):429 – 439.

Segurado, J., Lebensohn, R., LLorca, J., and Tome, C. (2012). Multiscale modeling of

plasticity based on embedding the viscoplastic self-consistent formulation in implicit

finite elements. International Journal of Plasticity, 28(1):124 – 140.

Segurado, J. and Llorca, J. (2002). A numerical approximation to the elastic proper-

ties of sphere-reinforced composites. Journal of the Mechanics and Physics of Solids,

50(10):2107 – 2121.

Segurado, J. and Llorca, J. (2013). Simulation of the deformation of polycrystalline nanos-

tructured Ti by computational homogenization. Computational Materials Science, 76:3

– 11.

Singh, A., Somekawa, H., and Mukai, T. (2007). Compressive strength and yield asym-

metry in extruded Mg-Zn-Ho alloys containing quasicrystal phase. Scripta Materialia,

56(11):935 – 938.

Sket, F., Enfedaque, A., Alton, C., Gonzalez, C., Molina-Aldareguia, J., and Llorca, J.

(2014). Automatic quantification of matrix cracking and fiber rotation by X-ray com-

puted tomography in shear-deformed carbon fiber-reinforced laminates. Composites Sci-

ence and Technology, 90:129 – 138.

Slutsky, L. J. and Garland, C. W. (1957). Elastic constants of magnesium from 4.2◦k to

300◦k. Phys. Rev., 107:972–976.

Song, Y., Shan, D., Chen, R., Zhang, F., and Han, E.-H. (2009). Biodegradable behaviors

of AZ31 magnesium alloy in simulated body fluid. Materials Science and Engineering:

C, 29(3):1039 – 1045.

111

Bibliography

Song, Y., Shan, D., and Han, E. (2008). Electrodeposition of hydroxyapatite coating on

AZ91D magnesium alloy for biomaterial application. Materials Letters, 62(17-18):3276

– 3279.

Souza, E., Peric, D., and Owens, D. (2008). Computational Methods for Plasticity: Theory

and Applications. Wiley.

Staiger, M. P., Pietak, A. M., Huadmai, J., and Dias, G. (2006). Magnesium and its alloys

as orthopedic biomaterials: A review. Biomaterials, 27(9):1728 – 1734.

Stanford, N. and Barnett, M. (2008). The origin of “rare earth” texture development

in extruded Mg-based alloys and its effect on tensile ductility. Materials Science and

Engineering: A, 496(1-2):399 – 408.

Stanford, N. and Barnett, M. (2013). Solute strengthening of prismatic slip, basal slip and

twinning in Mg and Mg�Zn binary alloys. International Journal of Plasticity, 47:165 –

181.

Stanford, N., Sabirov, I., Sha, G., La Fontaine, A., Ringer, S., and Barnett, M. (2010).

Effect of textscAl and gd solutes on the strain rate sensitivity of magnesium alloys.


Stanford, N., Sha, G., Xia, J., Ringer, S., and Barnett, M. (2011). Solute segregation and

texture modification in an extruded magnesium alloy containing gadolinium. Scripta

Materialia, 65(10):919 – 921.

Staroselsky, A. (1998). Crystal plasticity due to slip and twinning-Ph.D. Thesis. Mas-

sachusetts Institute of Technology.

Staroselsky, A. and Anand, L. (1998). Inelastic deformation of polycrystalline face centered

cubic materials by slip and twinning. Journal of the Mechanics and Physics of Solids,

46(4):671 – 696.

Taylor, G. (1938). Plastic strain in metals. Institute of metals, 62:307 – 324.

Tucker, J. C., Chan, L. H., Rohrer, G. S., Groeber, M. A., and Rollett, A. D. (2012). Com-

parison of grain size distributions in a Ni-based superalloy in three and two dimensions

using the Saltykov method. Scripta Materialia, 66(8):554 – 557.

112

Bibliography

Ulacia, I., Dudamell, N., Galvez, F., Yi, S., Perez-Prado, M., and Hurtado, I. (2010).

Mechanical behavior and microstructural evolution of a Mg AZ31 sheet at dynamic

strain rates. Acta Materialia, 58(8):2988 – 2998.

Vitek, V., Mrovec, M., and Bassani, J. (2004). Influence of non-glide stresses on plastic

flow: from atomistic to continuum modeling. Materials Science and Engineering: A,

365(1-2):31 – 37. Multiscale Materials Modelling.

Wang, H., Raeisinia, B., Wu, P., Agnew, S., and Tome, C. (2010). Evaluation of self-

consistent polycrystal plasticity models for magnesium alloy AZ31B sheet. International

Journal of Solids and Structures, 47(21):2905 – 2917.

Wang, S., Holm, E. A., Suni, J., Alvi, M. H., Kalu, P. N., and Rollett, A. D. (2011). Mod-

eling the recrystallized grain size in single phase materials. Acta Materialia, 59(10):3872

– 3882.

Yalcinkaya, T., Brekelmans, W. A. M., and Geers, M. G. D. (2008). BCC single crystal

plasticity modeling and its experimental identification. Modelling and Simulation in

Materials Science and Engineering, 16(8):085007.

Ye, J., Mishra, R. K., Sachdev, A. K., and Minor, A. M. (2011). In situ TEM compression

testing of Mg and Mg 0.2 wt 2%Ce single crystals. Scripta Materialia, 64(3):292 – 295.

Yi, S., Davies, C., Brokmeier, H., Bolmaro, R., Kainer, K., and Homeyer, J. (2006).

Deformation and texture evolution in AZ31 magnesium alloy during uniaxial loading.

Acta Materialia, 54(2):549 – 562.

Yoo, M. (1981). Slip, twinning, and fracture in hexagonal close-packed metals. Metallurgical

Transactions A, 12(3):409–418.

Yoo, M., Agnew, S., Morris, J., and Ho, K. (2001). Non-basal slip systems in HCP metals

and alloys: source mechanisms. Materials Science and Engineering: A, 319-321:87 – 92.

Yoshinaga, H., Obara, T., and Morozumi, S. (1973). Twinning deformation in magnesium

compressed along the C-axis. Materials Science and Engineering, 12(5-6):255 – 264.

Zhang, J. and Joshi, S. P. (2012). Phenomenological crystal plasticity modeling and de-

tailed micromechanical investigations of pure magnesium. Journal of the Mechanics and

Physics of Solids, 60(5):945 – 972.

113

Bibliography

Zhao, Z., Kuchnicki, S., Radovitzky, R., and no, A. C. (2007). Influence of in-grain mesh

resolution on the prediction of deformation textures in fcc polycrystals by crystal plas-

ticity FEM. Acta Materialia, 55(7):2361 – 2373.

Zhu, S. and Nie, J. (2004). Serrated flow and tensile properties of a Mg-Y-Nd alloy. Scripta

Materialia, 50(1):51 – 55.

114

List of Figures

1.1 Steering wheel of the US Toyota Camry (a), Faurecia’s front seat frame plat-

forms developed and produced for Nissan, General Motors and Volkswagen

(b), Boeing 737 thrust reverser (c), Toshiba Portege Z830� 104 with mag-

nesium alloy chassis (d), Bike with a frameset and wheels that are injection

metal molded in Mg (e) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 HCP crystallographic structure . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3 Plastic deformation modes in Mg . . . . . . . . . . . . . . . . . . . . . . . 6

1.4 Permanent deformation after by slip and twinning. . . . . . . . . . . . . . 7

1.5 EBSD image of Rh showing twins within the grains. [Kacher and Minor,

2014] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.6 Geometric configuration to determine the resolved shear stress τα on the slip

system characterized by the normal plane n and the slip direction s under

uniaxial loading σapplied. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.7 (Left) Typical pole figure of rolled Mg along ND direction. (Right) Section

A-A corresponding with the plane defined by RD-ND axes. [Zhang and

Joshi, 2012] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.8 (a) Orientation of the tensile axis with respect to the normal direction ND.

(b) Representative stress-strain curves of specimens tested at different angles

with respect to ND [Liu et al., 2011] . . . . . . . . . . . . . . . . . . . . . 11

1.9 Periodic microstructure and the corresponding RVE (a). Random polycrys-

tal microstructure and the corresponding RVE (taken from [Segurado and

Llorca, 2013]). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.10 Reference or undeformed configuration (B0) and current or deformed con-

figuration (B). Notation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

115

LIST OF FIGURES

1.11 Multiplicative decomposition of the total deformation gradient F into the

elastic, Fe, and plastic, Fp, components. . . . . . . . . . . . . . . . . . . . 15

1.12 Separation of scales between microscale and macroscale. . . . . . . . . . . . 17

1.13 VPSC assupmtion where the matrix-grain interaction is approximated by a

ellipsoidal grain (with its particular orientation) within a HEM . . . . . . . 18

1.14 Discretization of RVE of polycrystals. (a) Model with 1000 cubic voxels, in

which each one stands for a single crystal. (b) Model containing 100 crystals

discretized with 64000 voxels. (c) Model in which each crystal is represented

by a polyhedron obtained by means of a Voronoi tessellation. . . . . . . . . 22

1.15 Mg micropillar after compression in a direction at 45◦ from the basal plane

normal, showing slip along the basal plane. Courtesy of Yuan-Wei Edward

Chang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.1 Multiplicative decomposition indicating material point subdivision in parent

and twin phases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.2 CRSS evolution by hardening . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.3 Different reference systems used to characterize the planes in the hexagonal

lattice. Bravais (left) and orthogonal reference system (right) . . . . . . . . 33

2.4 Flow chart of the time discretization . . . . . . . . . . . . . . . . . . . . . 36

2.5 System reference rotation by Euler angles ϕ1, φ and ϕ2 . . . . . . . . . . . 39

2.6 Orientation of single crystal respect to a global system . . . . . . . . . . . 39

2.7 Rotation matrix corresponding with Rotations about axes z u and z′, see

Fig. 2.5. The letters c and s stand as the cosine and sine as well as the

sub-indices 1, 2 and 3 with the euler angles ϕ1, φ and ϕ2. . . . . . . . . . . 40

2.8 Different RVE of the polycrystal microstructure. (a)(b)(c)(d) Voxel repre-

sentation with 64, 216, 512 and 1000 cubic finite elements in which each

one stands for a grain respectively. (e)(f) Realistic RVE containing 584 and

300 crystals discretized with � 7 and 200 cubic finite elements per grain

respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

2.9 Flow chart of Levenberg-Marquardt optimization algorithm. . . . . . . . . 49

116

LIST OF FIGURES

3.1 Pole figures of the rolled AZ31 Mg alloy. (a) Experimental texture. (b)

Reduced equivalent initial texture with 512 orientations used as input to

create the RVE. The numbers in the legend stand for multiples of random

distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.2 Schematic of the loading directions for the mechanical tests of the rolled

plate of AZ31 Mg alloy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.3 Experimental true stress - true strain curves of the AZ31 Mg alloy along

different orientations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.4 Different RVE of the polycrystal microstructure for the optimization of the

AZ31 Mg alloy. (a) Voxel representation with 64 cubic finite elements in

which each one stands for a grain. (b) Voxel representation with 512 cubic

finite elements, one per grain. (c) Realistic RVE containing 512 crystals

discretized with � 7 cubic finite elements per grain. . . . . . . . . . . . . . 57

3.5 Evolution of the objective error function per point as a function of the

number of optimization iterations for different RVEs. . . . . . . . . . . . . 58

3.6 Results of the inverse optimization procedure using three stress-strain curves

(compression ND, tension ND and tension RD) as input. (a) Experimental

(solid lines) and numerical (broken lines with symbol) stress-strain curves

resulting from the optimization procedure. (b) Model prediction of the

tensile test in the RD-ND plane at 45◦ from both orientations. Solid lines

correspond to experimental results while broken lines with symbols stand

for the numerical simulations. The numerical results correspond to the RVE

with 512 crystals and � 7 elements per crystal. . . . . . . . . . . . . . . . 60

3.7 Relative contribution of each deformation mode to the plastic strain and

volume fraction of twinned material, f , as a function of the applied strain

in AZ31 Mg alloy. (a) Tension along ND. (b) Compression along ND. (c)

Tension along RD. (d) Tension along RD-ND plane at 45◦ from both orien-

tations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.8 Pole figures of the texture after 10% tensile strain along ND of the rolled

AZ31 Mg alloy. (a) Experimental results. (b) Computational homogeniza-

tion results obtained with the model with 512 orientations. The numbers in

the legend stand for multiples of random distribution. . . . . . . . . . . . . 64

117

LIST OF FIGURES

3.9 Results of the inverse optimization procedure using one stress-strain curves

(tension ND) as input. (a) Experimental (solid lines) and numerical (bro-

ken lines with symbol) stress-strain curves resulting from the optimization

procedure. (b) Model predictions of the compression test along ND and of

the tensile test along RD and in the RD-ND plane at 45◦ from both ori-

entations. Solid lines correspond to experimental results while broken lines

with symbols stand for the numerical simulations. The numerical results

correspond to the RVE with 512 crystals and � 7 elements per crystal. . . 66

3.10 Results of the inverse optimization procedure using two stress-strain curves

(tension along ND and RD) as input. (a) Experimental (solid lines) and

numerical (broken lines with symbol) stress-strain curves resulting from the

optimization procedure. (b) Model predictions of the compression test along

ND and of the tensile test in the RD-ND plane at 45◦ from both orientations.

Solid lines correspond to experimental results while broken lines with sym-

bols stand for the numerical simulations. The numerical results correspond

to the RVE with 512 crystals and � 7 elements per crystal. . . . . . . . . . 67

3.11 Optimized values of the initial CRSS (a), the saturation CRSS (b) and the

hardening modulus after the inverse analysis, for each of the deformation

modes. The input data used in the optimization process were the ND-T,

ND-C and RD-T curves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

3.12 Microstructure in the as-extruded condition (as shown in an optical mi-

crograph) and inverse pole figure showing the orientation of the extrusion

direction. (a) MN10 Mg alloy. (b) MN11 Mg alloy . . . . . . . . . . . . . . 71

3.13 Experimental true stress-strain curves of the RE-containing Mg alloys at

ambient temperature. (a) MN10. (b) MN11. . . . . . . . . . . . . . . . . . 72

3.14 Cubic RVE of the microstructure including 300 crystals discretized with 200

cubic finite elements per grain . . . . . . . . . . . . . . . . . . . . . . . . . 73

3.15 Experimental (solid lines) and simulated (broken lines) stress-strain curves

resulting from the optimization procedure in tension along ED, compression

along ED and compression at 90◦ from ED. (a) MN10 Mg alloy. (b) MN11

Mg alloy. (c) Experimental (solid lines) and predicted (broken lines) stress-

strain curves corresponding to both alloys tested in compression at 45◦ with

respect to ED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

118

LIST OF FIGURES

3.16 Experimental and simulated inverse pole figures showing the orientation of

the compression direction of the MN10 and MN11 Mg alloys after compres-

sion along ED. The numbers in the legend stand for multiples of random

distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

3.17 Optical micrographs showing the grain structure of the MN11 Mg alloy per-

pendicular to the ED. (a) As-extruded bar. (b) Grip section of the specimens

tested in tension along ED at -175◦C. (c) Idem at 50◦C, (d) Idem at 150◦C.

(e) Idem at 250◦C. (f) Idem at 300◦C. The average grain sizes are included

as insets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

3.18 Transmission electron microscopy micrographs of the grip section of the

specimens tested in tension at 10−3s−1 and different temperatures. (a) -

175◦C and (b-d) 250◦C. Micrographs are perpendicular to the ED. . . . . . 81

3.19 Inverse pole figures illustrating the orientation of the ED. (a) As-extruded

bar. (b) Grip section of the specimens tested in tension along ED at -175◦C.

(c) Idem at 50◦C. (d) Idem at 150◦C. (e) Idem at -250◦C. (f) Idem at -300◦C.

The numbers in the legend stand for multiples of random distribution. . . . 82

3.20 Experimental true stress-strain curves of the MN11 Mg alloy at different

temperatures. (a) Compression along ED. (b) Tension along ED. . . . . . . 83

3.21 Evolution of the yield stress in tension and compression along the ED with

temperature for the MN11 Mg alloy. . . . . . . . . . . . . . . . . . . . . . 84

3.22 Electron backscatter diffraction inverse pole figure maps in the ED of spec-

imens compressed up to 5% engineering strain at 10−3s−1 at different tem-

peratures. (a) 50◦C and (b) 250◦C. The non-indexed points are shown as

black pixels. The boundaries having a misorientation of 86◦ (� 5◦) have

been depicted as white lines. The compression axis is horizontal. . . . . . . 86

3.23 Inverse pole figures obtained by X-ray diffraction showing the orientation of

the ED in specimens deformed under different conditions. (a) 50◦C under

compression up to 30% engineering strain. (b) 50◦C under tension up to

failure. (c) 250◦C under compression up to 40% engineering strain. (d)

250◦C under tension up to failure. The numbers in the legend indicate

multiples of random distribution. . . . . . . . . . . . . . . . . . . . . . . . 87

119

LIST OF FIGURES

3.24 Experimental (solid lines) and simulated (broken lines) stress-strain curves

in tension and compression along ED in MN11 Mg alloy at different tem-

peratures.(a) -175◦C. (b) 50◦C. (c) 150◦C. (d) 250◦C, (e) 300◦C. . . . . . . 92

3.25 Evolution of the initial CRSS, τ0,c, with temperature for each slip mode and

twinning of MN11 Mg alloy according to the inverse optimization model. . 93

3.26 Relative contribution of each deformation mode to the plastic strain as a

function of temperature and loading (tension or compression along the ED)

in MN11 Mg alloy. (a) -175◦C, tension. (b) -175◦C, compression. (c) -50◦C,

tension. (d) 50◦C, compression. (e) 250◦C, tension. (f) 250◦C, compression. 94

120

List of Tables

1.1 Elastic constants (in GPa) of Mg single crystal at 300K [Slutsky and Gar-

land, 1957]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.2 Values of the initial CRSS for different slips modes and tensile twinning in

AZ31 Mg alloy predicted by fitting experimental results on polycrystals with

simulations based on mean-field methods or computational homogenization. 25

2.1 Deformation systems considered. Plane normals n and slip directions s are

expressed both in the Bravais coordinated system (a1, a2, a3, c) (sub-index

brav) and in the orthogonal system (e1, e2 and e3) (sub-index ort) . . . . . . 34

2.2 Internal variables (STATEV) saved at each point of convergence and for

each Gauss point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.1 Chemical composition of the AZ31 alloy in wt.%. . . . . . . . . . . . . . . 51

3.2 Average values of the maximum Schmid factors for different deformation

modes in the polycrystalline AZ31 Mg alloy. . . . . . . . . . . . . . . . . . 55

3.3 Optimum values of the parameters that define the mechanical behavior of

each slip mode and extension twinning in the AZ31 Mg alloy as a function

of the RVE used in the optimization process. Magnitudes are expressed in

MPa. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59


each slip mode and extension twinning in the AZ31 Mg alloy as a func-

tion of the input stress-strain curves used in the optimization procedure.

Magnitudes are expressed in MPa. . . . . . . . . . . . . . . . . . . . . . . . 65

3.5 Chemical composition of MN10 and MN11 Mg alloys (in wt.%). . . . . . . 70

121

LIST OF TABLES

3.6 Comparison of the initial CRSSs (τ0,c) obtained by inverse optimization for

the MN10 and MN11 alloys with those measured in pure Mg single crystals

[Zhang and Joshi, 2012] . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76


each slip mode and extension twinning in MN11 Mg alloy as a function of

temperature. Magnitudes are expressed in MPa. . . . . . . . . . . . . . . . 91

122

Appendix BPersonal contributions

Papers in journals

� Vicente Herrera Solaz; Fco. Javier Llorca Martinez; Ebubekir Dogan; Ibrahim Kara-

man; Javier Segurado Escudero. “An inverse optimization strategy to determine

single crystal mechanical behavior from polycrystal tests: Application to AZ31 Mg

alloy”. International Journal of Plasticity. 57, pp. 1�15. 02/2014. ISSN 0749�6419.

Impact factor= 4.35

� Vicente Herrera Solaz; Paloma Hidalgo Manrique; Maria Teresa Perez Prado; Diet-

mar Letzig; Fco. Javier Llorca Martinez; Javier Segurado Escudero. “Effect of rare

earth additions on the critical resolved shear stresses of magnesium alloys”. Materials

Letters. 128, pp. 199� 203. 04/2014. ISSN 0167� 577X. Impact factor= 2.22

� Vicente Herrera Solaz; Fco. Javier Llorca Martinez; Javier Segurado Escudero. De-

terminacion de propiedades de monocristales a partir de ensayos mecanicos en poli-

cristales: Aplicacion a aleaciones de Magnesio. Anales de Mecanica de la fractura.

04/2014.

� Paloma Hidalgo Manrique; Vicente Herrera Solaz; Javier Segurado Escudero; Fco.

Javier Llorca Martinez; Francisco Galvez Diaz-Rubio; O.A. Ruano; Maria Teresa

Perez Prado. “Origin of the reversed yield asymmetry in Mg-rare earth alloys at high

temperature”. Paper submitted for publication to Acta Materialia. ISSN 1359�6454.

Impact factor= 3.9

123

Appendix B. Personal contributions

� Vicente Herrera Solaz; Javier Segurado Escudero; Fco. Javier Llorca Martinez.“On

the robustness of an inverse optimization approach based on Levenberg-Marquardt

method for the mechanical behavior of polycrystals”. Paper submitted for publica-

tion.

Congresses

� Stochastic and multiscale inverse problems. “An inverse optimization strategy to

determine single crystal mechanical behavior from polycrystal tests: application to

Mg alloys”. 02/10/2014. Paris-FRANCE.

� 24th International Workshop on Computational Micromechanics of Materials (IWCMM24).

“Effect of Temperature on the critical resolved shear stresses of MN11 Magnesium

alloy”. 01/10/2014. Getafe (Madrid)-SPAIN.

� 17th U.S. National Congress on Theoretical and Applied Mechanics. “An inverse

optimization strategy to determine single crystal mechanical behavior from polycrys-

tal tests: application to Mg alloys”. 15/06/2014. Michigan State University, East

Lansing (MI)-USA.

� 3rd International Workshop on Physics Based Material Models and Experimental

Observations. “An inverse optimization strategy to determine single crystal mechan-

ical behavior from polycrystal tests: application to Mg alloys (Poster)”. 02/06/2014.

Cesme/Izmir-TURKEY

� XXXI Encuentro del grupo espanol de fractura. “Determinacion de propiedades de

monocristales a partir de ensayos mecanicos en policristales: Aplicacion a aleaciones

de Magnesio”. 02/04/2014. San Lorenzo del Escorial (Madrid)-SPAIN.

� Magnesium Workshop. “An International Workshop on Processing-Microstructure-

Mechanical Property of Magnesium Alloys”. Crystal Plasticity Modeling of Magne-

sium Alloys. 21/05/2013. Getafe (Madrid)-SPAIN.

Stays abroad

� Michigan State University, East Lansing (MI)-USA. From 1/08/2013 to 31/10/2013.

124

Microstructure-based numerical modeling of the mechanical...

Documents

Transcript of Microstructure-based numerical modeling of the mechanical...