tomographic investigation of creep and creep damage under inhomogeneous loading conditions ·...

Tomographic Investigation of Creep and Creep Damage under Inhomogeneous Loading Conditions

Dissertation zur

Erlangung des Grades Doktor-Ingenieur

der Fakultät für Maschinenbau

der Ruhr-Universität Bochum

von

Federico Iván Sket aus Cruz del Eje, Argentinien

Bochum 2010

Dissertation eingereicht am: 13.01.2010 Tag der mündlichen Prüfung: 21.05.2010 Erster Referent: Prof. Dr.-Ing. Anke Rita Pyzalla Zweiter Referent: Prof. Dr. András Borbély

I

TOMOGRAPHIC INVESTIGATION OF CREEP AND CREEP

DAMAGE UNDER INHOMOGENOUS LOADING

CONDITIONS

Dipl.-Ing. Federico Iván Sket

ABSTRACT

Engineering components are usually subjected to inhomogeneous loading conditions,

when material behaviour cannot be easily predicted as a simple superposition of different

effects characteristic for uniform loading. To better understand complex material behaviour

under inhomogeneous loading adequate investigation methods are required. This thesis

investigates the applicability of X-ray microtomography for the study of creep and creep

damage under inhomogeneous loading. The thesis has two main parts describing effects

caused by inhomogeneous temperature and inhomogeneous stress distribution.

Exploiting the nondestructive nature of X-ray tomography the first part of the thesis

shows that in situ tomography during creep is well applicable to characterize changes in

specimen’s shape. A new method for the evaluation of the apparent activation energy of

steady-state creep is proposed, which is based on in situ monitoring the local cross-section of

a cylindrical specimen subjected to uniaxial load and linear temperature distribution. It is

shown that microtomography acting as a three dimensional extensometer enables the

evaluation of local strain-rates. Good agreement between strain distributions evaluated from

the real tomographic measurement and finite element models was obtained. Activation

energies obtained with the new method for stainless steel agree within an error of 5% with

values obtained according to the classical procedure.

The second part of the thesis describes damage distribution in a notched hollow cylinder

made of E911 steel and crept for 26,000 h under multi-axial stress. A detailed characterization

of creep cavities is presented, comprising relevant statistical information about the size,

shape, orientation, and cavity density along the notch radius. Using a proper shape descriptor,

a separation of non-coalesced and coalesced cavities could be made. The analysis of non-

coalesced cavities in terms of general power-law functions describing nucleation and growth

led to the conclusion that cavity growth in E911 steel is dominated by the constrained

II

diffusion mechanism. The cross-correlation analysis between damage and different stress

parameter distributions along the notch radius enabled emphasizing the important influence

on damage of stress triaxiality, maximum principal stress and the von Mises equivalent stress.

III

ACKNOWLEGMENTS

The present work was carried out during my three and half years activities as a scientific

co-worker in the Department of Material Diagnostics and Steel Technology of the Max-

Planck-Institut für Eisenforschung GmbH (MPIE) in Düsseldorf and at Helmholtz-Zentrum

Berlin GmbH (HZB), Berlin, Germany.

First of all I would like to express my thanks to my advisor Prof. Dr.-Ing. Anke

Kaysser-Pyzalla, now at HZB, who provided me constant guidance, fruitful suggestions and

discussions and the financial support that permitted me to carry out this work at the MPIE and

the HZB.

I am also very grateful to Prof. Dr. András Borbély, now at "Ecole des Mines de Saint

Etienne" for guiding me through the whole process of learning. As an expert in Material

Science, he gave me insightful comments and challenged me to refine thoughts. He made

possible this thesis with his guidance and the fruitful discussions.

I would also like to thank to Prof. Dr. Karl Maile, at MPA Stuttgart, for the

colaboration, discussions and creep samples provided for evaluation in this thesis.

I would also like to express my thanks to my colleagues Krzystof Dzieciol and Augusta

Isaac who have been very active discussion partners during this thesis and the synergy

generated during this time made possible the completion of this thesis. I´m very grateful to

have met you guys!

My special thank to Gerhard Bialkowski whose help in the experimental part was

essential for the development of this thesis. His good mood and efficiency was very important

to create a very nice working environment.

The friendly supportive atmosphere inherent to the whole working group contributed to

the final outcome of my studies, not only during the working hours at the institute but also

during the free time. To all of them, Dimas Souza, Rodrigo Coelho, Pedro Brito, Marcin

Moscicki, Leonardo Agudo, David Rojas, Orlando Prat, Haroldo Pinto, Carla Barbatti, Pedro

Silva, Jose García, Mauro Martin, Maitena Dumont, Aleksander Kostka, Derek Leach, Hauke

Springer, Lais Mujica, Rosario Maccio, Fernando and Andres Lasagni, Claudia Juricic and

Adelheid Adrian I am truly grateful.

I greatly acknowledge the staff of the ID15 and ID19 at the European Synchrotron

Radiation Facilities, specially Dr. T. Buslaps, Dr. di Michiel, Dr. Boller and Dr. Tafforeau,

for their helpful assistance during the tomography experiments with synchrotron radiation.

IV

My special appreciation goes to my parents, Hugo and Corina, my brothers Hugo and

Germán and my sisters in law Andrea and Luciana who were always there for me. Their love

and encouragement were the driving forces that kept me going in the difficult moments. My

girlfriend, María, who became a very important part of my life and made it easier and of

course I cannot forget my nephew and godson, Huguito, who made me part of his life even

being at 14,000 km away from me.

V

TABLE OF CONTENTS Chapter 1 – Introduction…………..………………………………...………………….

1.1 Background………………...…………………………………………………...

1.2 Preliminary works……………………………………………...……………….

1

1

3

Chapter II – Creep deformation, State-of-the-art……………………………………..

2.1 Creep of metals and alloys………………………………………………….......

2.2 Power-Law Creep…………………….…………………………………...........

2.3 Rate-controlling mechanism……………………………………………………

2.4 Methods for the determination of Activation Energy of Creep ……..…...……..

2.5 Creep damage under multiaxial stress………………………………………….

2.5.1 Experimental techniques for damage evaluation………………………….

2.5.2 Development of creep resistant steels……………………………………..

2.5.3 Diffusion models of cavity growth………………………………………...

2.5.3.1 Grain boundary diffusion controlled growth…………………………

2.5.3.2 Surface diffusion controlled growth………………………………….

2.5.3.3 Constrained diffusional cavity growth……………………………….

2.5.4 The cavity size distribution function……………………………………...

2.5.5 Prediction of creep rupture time (phenomenological approach)…………..

4

4

6

9

11

17

17

19

22

23

25

26

30

32

Chapter III – Experimental details……………………………………………………..

3.1 Uniaxial in situ experiments……………………………………………………

3.1.1 Materials and specimen geometry adapted to tomographic measurement...

3.1.2 Creep device for in situ experiments and testing conditions…………...…

3.1.3 Creep tests with nearly constant load or constant stress…………………..

3.2 Multi-axial creep experiments on E911 steel…………………………………...

3.2.1 Materials and specimen geometry adapted to tomographic measurement

conditions………………………………………………………………….

3.3 Synchrotron X-ray microtomography…………………………………………..

3.3.1 X-ray tomography: Principles…………………………………………...

3.4 Image processing………………………………………………………………..

3.4.1 Reconstruction and pre-processing………………………………………..

3.4.2 Identification of cavities…………………………………………………...

3.4.3 Image correlation for evaluation of tomographic data…………………….

37

37

37

38

40

42

42

43

44

46

46

48

50

VI

Chapter IV – Results………………………………………………………………….....

4.1 Method for evaluation of apparent activation energy of creep, aQ ……………

4.1.1 Temperature distribution calibration………………………………………

4.1.2 Image correlation for evaluation of tomographic data…………………….

4.1.3 Method for evaluation on samples with constant cross sectionaQ ......…...

4.1.3.1 In situ creep curve of brass (samples A)...............................................

4.1.3.2 aQ values according to literature and tomographic method………….

4.1.3.3 Damage development in brass (samples A)……………...…...……...

4.1.4 Method for evaluation on samples with varying cross sectionaQ ………..

4.1.4.1 aQ values according to classical and tomographic methods…………

4.1.4.2 In situ creep curves of steel (samples B and C)....…………………..

4.2 Microtomographic investigation of damage in E911 steel after long term creep

4.2.1 Conventional damage evaluation by OM………………………………….

4.2.2 Tomographic evaluation of damage……………………………………….

52

52

55

56

58

59

60

62

64

64

66

70

71

73

Chapter V – Discussion……………………………………………………...…………..

5.1 Evaluation of apparent activation energy of creep, aQ ……...................………

5.1.1 Comparison of strain distribution in the real sample and FE models…......

5.1.2 Error estimation……………………………………………………………

5.2 Damage investigation in E911 steel…………...…………………...…………...

81

81

81

84

85

Chapter VI – Conclusions and perspectives………………………............……….….. 92

Chapter VII – References……...…………..……………………………….……….….. 94

Curriculum vitae……...…………..………………………………...…….……...….….. 105

I. Introduction 1

CHAPTER I

INTRODUCTION 1.1 Background

Creep deformation of metals at temperatures higher than about 1/3rd of the melting

temperature is governed by several thermally activated processes enhancing dislocation

motion and recovery, diffusion of vacancies as well as dynamic recrystallization [

mT

1]. These

mechanisms influence mainly stage II of creep, where specimens under constant stress deform

usually at nearly constant strain-rate. The relationship between this steady-state strain-rate and

stress in pure metals and Class M alloys is often described by a power-law function [1,2]. The

key parameters of this phenomenological description are the stress exponent, , and the

apparent activation energy for steady-state creep, , which once evaluated give good hints

about the main creep mechanism. It is the aim of the first part of this work to present a new

evaluation method of based on in situ microtomography.

n

aQ

aQ

Due to the economical importance of creep at high temperatures much attention has

been given to the mechanisms of intergranular cavities growth. Creep rupture by cavity

growth and coalescence is an important failure mechanism for high temperature components

for example in power plants. Knowledge of damage development during high temperature

creep of materials is extremely important for the prediction of service lifetimes of many

engineering components. The subject was extensively studied both theoretically [3,4,5,6] and

experimentally [7,8,9]. Review articles [10,11,12] and books e.g. [13] are presenting the main

achievements in the field, but it is nevertheless quite difficult to compare existing

experimental data with model predictions. The reason is twofold:

a) There are several damage mechanisms acting simultaneously such as void nucleation,

void growth by diffusion, by plasticity or by coupling between them. Due to this complexity

global damage characterization techniques like Archimedean densitometry, optical/scanning

electron microscopy or small angle X-ray/neutron scattering [7,14,15] are usually inadequate

to conclude on the validity of a given damage model. Usually difficult assumptions have to be

made about the fraction and coupling among different mechanisms. The problem is further

complicated by the absence or inaccurate knowledge of creep material paramters, especially

for engineering materials, required for a quantitative check of the models.

I. Introduction 2

b) Due to the lack of adequate experimental techniques generally no complete three-

dimensional (3D) evaluation of microscopic damage was performed. The destructive nature of

some techniques (e.g. optical or electron microscopy) hinders a univocal characterization of

damage evolution, since data obtained from two-dimensional (2D) sections are inherently

affected by statistical fluctuations as well as by errors related to the 3D connectivity of larger

voids.

Under multiaxial and locally varying stress creep damage develops inhomogenously

and a quantitative assessment is very difficult by conventional metallographic techniques.

Creep lifetime and failure mode under multiaxial stress conditions can differ significantly

from the rules established for uniaxial loading. For example Cocks and Ashby [5] have

predicted that cavity growth is highly accelerated by high positive stress triaxiality. There are

numerous applications in which engineering components are subjected to inhomogeneous

loading conditions of temperature and stress. In a steam power plant for example the most

exposed components are usually the boiler tubes, headers and turbines, where these materials

are subjected to inhomogeneous loading. In case of a hollow cylinder for example the

multiaxiality of the stress state is imposed by internal pressure, additional longitudinal force

as well as sample geometry. The assessment of the lifetime and failure modes of such

industrial components needs the evaluation of damage on samples subjected to similar

conditions as in service. The stress parameters usually used for the assessment of rupture life

are, however, averages over the local values in the sample. Multiaxial loading of large

specimens leads, however, to inhomogenous damage indicating that local parameters should

be considered for appropriate understanding of the results. The high spatial resolution

achievable with synchrotron microtomography (of about 0.33 μm) plus the possibility to

reconstruct representative volumes makes this technique very attractive for the 3D

characterization of inhomogeneous damage distribution.

It is the aim of the second part of this thesis to assess the damage in a notched hollow

cylinder of E911 steel, which was subjected to about 26,000 h of creep of similar loads as in

service. The 3D shape, orientation and size distribution of single cavities will be described. A

comparison of local damage distributions with the distribution of selected stress parameters

(knowing to influence damage) as obtained from FE simulation will be performed. The

tomographic results will also be compared with those of conventional metallography.

I. Introduction 3

1.2 Preliminary works

This thesis was preceded by the following publications:

(1). F. Sket, A. Isaac, K. Dzieciol, A. Borbély, K. Maile, A.R. Pyzalla, Microtomographic

investigation of damage in E911 steel after long term creep, Mat. Sci. Eng. A, to be

submitted.

(2). F. Sket, K. Dzieciol, A. Isaac, A. Borbély, A.R. Pyzalla, Tomographic method for

evaluation of apparent activation energy of steady-state creep, Mat. Sci. Eng. A.

(2009) accepted for publication.

(3). F. Sket, A. Isaac, K. Dzieciol, G. Sauthoff, A. Borbély, A.R. Pyzalla, In situ

tomographic investigation of brass during high-temperature creep, Scr. Mater. 59

(2008) 558-561.

(4). K. Dzieciol, A. Isaac, F. Sket, A. Borbély, A.R. Pyzalla, Application of correlation

techniques to creep damage studies, Collected Proceedings TMS Conference, San

Francisco 15-19 February (2009), Characterization of Minerals, Metals and Materials,

pp. 15-22.

(5). A. Isaac, F. Sket, W. Reimers, B. Camin, G. Sauthoff, A.R. Pyzalla, In-situ 3D

quantification of the evolution of creep cavity size, shape, and spatial orientation using

synchrotron X-ray tomography, Mat. Sci. Eng. A 478 (1-2) (2008) 108-118.

(6). A. Isaac, F. Sket, A. Borbély, G. Sauthoff, A.R. Pyzalla, Study of cavity evolution

during creep by synchrotron microtomography using a volume correlation method,

Praktische Metallographie/Practical Metallography 45 (5) (2008) 242-245.

(7). A. Isaac, K. Dzieciol, F. Sket, M. di Michiel, T. Buslaps, A. Borbély, A.R. Pyzalla,

Investigation of creep cavity coalescence in brass by in-situ synchrotron X-ray

microtomography, In: S.R. Stock, Editor, Developments in X-Ray tomography VI,

Proc. of SPIE, 7078 (2008) pp. J1-J10.

II. State of the Art 4

CHAPTER II

CREEP DEFORMATION

STATE-OF-THE-ART

2.1 Creep of metals and alloys

The strain response of a body to the applied stress varies with the magnitude and state

of stress, temperature, and strain-rate. At homologous temperatures ( , where Tm is the

absolute melting point) above 0.3 it is a reasonable and widely used idealization to consider

the elastic-plastic behavior of metals as time-dependent. When a material undergoes

continuous deformation under constant load or stress it is said to creep and this may include

elastic, viscous and plastic deformations. The plasticity under these conditions is described in

Figure 2.1 for constant load. The instantaneous strain

m/ TT

0ε (Figure 2.1 (a), (b), and (c)) is

obtained immediately upon loading and exhibits characteristics of plastic deformation.

In Figure 2.1 (a) three regions are observed. Stage I, or primary creep, which denotes

the portion where the creep-rate (plastic strain-rate) dtd /εε =& is changing with increasing

plastic strain or time (until 1ε and time 1t Figure 2.1 (a) and (d)). The decreasing creep-rate

in the primary creep stage has been attributed to strain hardening and to the decrease of the

density of mobile dislocations.

in

Stage II, where the strain-rate has a constant value over a range of strain. This

phenomenon is also termed secondary or steady-state creep and is commonly attributed to a

state of balance between the rate of generation of dislocations contributing to hardening and

the rate of recovery contributing to softening. Beyond 2ε (Figure 2.1 (a)) or (Figure 2.1

(b)) cavitation and/or cracking increase the apparent strain-rate continuously until rupture

occurs at the strain

2t

rε and rupture time . rt

The region with increasing creep-rate is called Stage III or tertiary creep and is followed

by fracture. The increase in creep-rate with time in the tertiary creep stage can be a

consequence of increasing stress (due to necking of the specimen) or of microstructure

evolution (dynamic recovery, dynamic recrystallization, coarsening of precipitates and other

phenomena, which cause softening and result in a decrease in resistance to creep) and damage

evolution (development of creep voids and cracks, often along grain boundaries) during creep.


Figure 2.1. (a), (b) and (c) Creep curves of engineering materials under constant tensile load

and constant temperature and (d), (e) and (f) their creep-rate curves as a function of time [16].

The type of creep curve depicted in Figure 2.1 (a) and (d) is not always observed during

creep of metals and alloys. Whether all creep stages are observed in any test depends on

temperature, stress, and the duration of the test. Under certain conditions, the tertiary creep

stage begins immediately after the primary creep and predominates over the other stages [13],

as shown in Figure 2.1 (b) and (e). In this case, the steady-state creep-rate ssε& is defined as

minimum creep-rate minε& . The minimum creep-rate can be also explained by the process

where hardening in the primary stage is balanced by softening in the tertiary stage. In many

cases, there is substantially no steady-state stage suggesting that there is no dynamic

microstructural equilibrium during creep. At low homologous temperatures ( ), often

less than 0.3, only primary creep appears. At these temperatures diffusion is not important and

often the strain is well below 1% and it doesn’t lead to final fracture (Figure 2.1 (c) and (f)).

This deformation process is designated as logarithmic creep.

m/TT


2.2 Power-Law Creep

The creep-rate at constant stress usually increases exponentially with temperature

[17,18] following an Arrhenius-type law. The creep-rate also depends on the stress according

to a power law, i.e. , is the stress exponent. In pure metals and Class M alloys

there is an established, largely phenomenological relationship between the steady-state strain-

rate and stress which is often described by a power-law function [

nσε ∝ss& n

1,2]:

2.1) ⎟⎠⎞

⎜⎝⎛

⋅−⋅⎟⎟

⎠

⎞⎜⎜⎝

⎛⋅=

TRQ

TGC

n

ass exp

)(σε&

where is the apparent activation energy for creep, is the gas constant, the pre-

exponential factor C - which comprises all other structure-dependent parameters - is assumed

to be constant [

aQ R

19], )(/ TGσ is the shear modulus normalized stress (some authors are using

the Young’s modulus ), is the stress exponent, and )(TE n T the absolute temperature. The

values of are often found to be about that of lattice self-diffusion, . The stress

exponent, , is in the range of 4-7 and it is constant over a relatively large range of

temperatures and strain-rates. Eq.

aQ

n

SDQ

2.1 is often referred to as Norton’s law, power-law or five-

power-law (due to the typical value of the stress exponent). Steady-state creep is often

emphasized over primary or tertiary creep due to the relatively large fraction of creep life

within this regime. The concept of five-power-law creep, however, must be described in the

context of other (usually commercially less important) types of creep (e.g. Nabarro-Herring,

Harper-Dorn, Coble, low-temperature creep (Power Law Breakdown - PLB) as well as three-

power viscous glide creep), as illustrated in Figure 2.2.

The importance of steady-state is evidenced by the empirical relationship suggested by

Monkman and Grant [20], which shows that the overall time to fracture is controlled by the

steady-state creep-rate (

ft

ssε& )

2.2) MGssf Ct m =⋅ε&

where is a constant depending on total elongation during creep and is a constant

often nearly equal to 1. The Monkman-Grant relationship is a surprisingly simple observation

MGC m


which has been experimentally confirmed not only for simple metals and alloys [21,22,23]

but also for a number of engineering creep-resistant steels [16,24]. The wide range of

conditions for which the relation has been confirmed demonstrates that the Monkman-Grant

constant ( ) is independent of stress and temperature [MGC 21,22,23] and its approximate

constancy places constraints on creep cavitation theories.

Figure 2.2. Ashby deformation map of pure copper [25] with a grain size of 0.1 mm,

including power law breakdown.

The experimentally proven temperature dependence of creep-rate described by Eq. 2.1

reflects the fact that creep involves thermally activated processes and thermally activated

micromechanisms operating on the atomic scale. The first step in determining the

micromechanisms controlling the creep-rate is to obtain the activation energy. If one of these

mechanisms dominates, the activation energy of creep will be identical with the activation

energy of this mechanism. Creep, however, is a very complex phenomenon, and under certain

conditions several mechanisms may be operating simultaneously. The physical interpretation

of the activation energy determined under these conditions requires further consideration.

Here, two cases can be distinguished [26,27].


The first one, when the processes are independent or parallel-concurrent, they are active

simultaneously and contribute with a strain iε , the strain-rates are additive and the Arrhenius

plot is the sum of exponential functions, one for each process (given by Eq. 2.1). However,

the frequent case is that no more than two processes contribute to the creep-rate to a

comparable extent. The mechanisms usually have different activation energies and so

contribute significantly to the creep-rate only over a relatively narrow temperature interval. In

any other temperature range, the fastest process will dominates creep.

The second case occurs when the processes are dependent and act in series (often called

series-sequential). In case of two processes acting in series-sequential, the second process

cannot operate until the first one has taken place and vice versa (mutually accommodating

processes). The time necessary for occurrence of the first and second process are additive.

Then, in any given temperature range, the slower process will control the creep.

Figure 2.3. Arrhenius plot for a) Parallel-concurrent processes and b) Series-sequential

processes [28].

A knowledge of the activation energy of creep, , and the stress exponent, n , give

indications about the main creep process and the atomic mechanisms controlling the creep-

rate under a given external condition (temperature and applied stress). In the following

sections, the rate-controlling mechanisms are briefly described and then a concisely

description of the state-of-the-art of the usual evaluation procedures for the activation energy

determination is given which will lead to the results of the first part of the thesis in which a

new method for activation energy determination based on tomographic measurements is

presented.

aQ


2.3 Rate-controlling mechanism

Several mechanisms can be responsible for creep; the rate-controlling mechanism

depends both on the stress and on the temperature. For the temperature range higher than

about the creep mechanisms can be divided into two major groups: m4.0 T⋅

• Grain boundary mechanisms: in which grain boundary and, therefore, grain size,

play a major role.

• Lattice mechanisms: which occur independently of grain boundaries.

Dislocation glide is the most important mechanism of plastic deformation. In the course

of plastic deformation by dislocation glide alone, the dislocation density increases, leading to

an increasing flow stress with the strain. This mechanism takes place at relatively low

temperatures and high strain-rates (in the stress range 210 ). Below the ideal shear

strength, flow by the conservative motion of dislocations (glide) is possible provided an

adequate number of independent slip systems are available. This motion is almost always

obstacle-limited, i.e. by the interaction of mobile dislocations with other dislocations, with

solute precipitates, with grain boundaries, or with the friction of the lattice itself which

determines the rate of flow and (at a given rate) the yield stress. Dislocation glide is a kinetic

process, and the strain-rate produced by the average velocity of mobile dislocations, m

/ −≥Gσ

ρ ,

moving through a field of obstacles is almost entirely determined by their waiting time at

obstacles [29]. In the most interesting range of stress, the mobility of the dislocations and thus

their average velocity is determined by the rate at which dislocation segments are thermally

activated through, or around, obstacles. Two classes of obstacles can be defined: discrete

obstacles which can be bypassed or cut by a moving dislocation, depending on the

temperature and stress; and extended obstacles, diffuse barriers to dislocation motion (e.g.

lattice friction or a concentrated solid solution).

When the homologous temperature at which plastic deformation take place is higher

than 0.4, dynamic recovery starts to play an important role. Orowan proposed that creep is a

balance between the work-hardening (due to plastic deformation) and recovery (due to

exposure at high temperatures). The recovery during creep can occur by various mechanisms.

The most important of them involves non-conservative motion of dislocations (e.g. climb) and

annihilation of dislocations, and therefore depends on diffusion which can occur either via the

lattice (higher homologous temperature) or via dislocation cores (lower homologous

temperature). The non-conservative motion of dislocations (if the dislocations are properly

arranged) can in itself represent a mechanism of plastic deformation [30,31]. In the stress


range , creep tends to occur by dislocation glide, aided by vacancy

diffusion (when an obstacle is to be overcome).

24 10/10 −− ≤≤ Gσ

Another deformation mechanism, which usually participates in polycrystals at

homologous temperatures higher than 0.4, is grain boundary sliding (GBS). Under the action

of applied stress, in the diffusional creep regime, GBS is accommodated by atoms transported

from boundaries subjected to compressive stress, to those of subjected to tensile stress. This

leads to changes in shape of individual grains. At higher stresses, in the dislocation creep

regime, grain boundary sliding is accommodated by dislocation glide. Dislocations move by

glide and climb, so as to remove or supply matter to the part of boundary which needs it. If

GBS is not accommodated by either diffusion of vacancies or dislocation glide, an extreme

heterogeneity of plastic deformation occurs. This leads to the reduction of polyrcystal

compatibility by formation of voids on grain boundaries.

Diffusion creep occurs for (this value to a certain extent depends on the

metal). Two mechanisms are considered important in this region. The first one takes place at

high temperatures (about 0.7·Tm) by diffusion of vacancies via lattice (Nabarro-Herring creep)

producing an increase in the length of the grain along the direction of the applied (tensile)

stress. The second one is based on diffusion along the grain boundaries (instead of bulk) and

it is called Coble creep. It occurs at lower temperatures than Nabarro-Herring creep and

results in sliding of the grain boundaries. The diffusional creep-rate is inversely proportional

to the second power of mean grain diameter when the diffusion mainly occurs via the lattice

(Nabarro-Herring creep), and to the third power of mean grain diameter when it occurs

through grain boundaries (Coble creep). Harper and Dorn observed another type of

diffusional creep in aluminium [

410/ −≤Gσ

32], which occurred at high temperatures and low stresses,

and the creep-rates were over 1000 times greater than those predicted by Nabarro-Herring

creep. They conclude that creep occurred exclusively by dislocation climb. A significant

contribution of Harper-Dorn creep occurs for large grain sizes (> 400µm).

When the integrity of the crystal is damaged, i.e. the incompatibility created by the

grain boundary sliding is not accommodated by any of the above mentioned mechanisms, the

nucleation and especially the growth of cavities or cracks appear at the grain boundaries

which lie mainly perpendicular to the tensile axis [3,33] (proceeding by grain boundary

diffusion, intragranular deformation, grain boundary sliding or by various combination of

these processes). Thus, the cavities naturally contribute to the measured strain.

In the light of the many possible mechanisms that lead to creep rupture, it follows that

the plastic strain as measured in a tensile creep test generally consists of several components:


2.3) vdgbndg εεεεεε ++++=

where dgε , nε , gbε , dε , vε are the strain caused by dislocation glide, non-conservative

motion of dislocations, grain boundary sliding, stress directed diffusion of vacancies and by

intercrystalline void nucleation and growth, respectively. Not all processes operating during

creep as mentioned in Eq. 2.3 are independent of each other, as frequently assumed. An idea

about the possible mechanism dominating creep under a given condition is provided by the

deformation mechanisms maps [25] on the assumption that all deformation mechanisms

considered are mutually independent, and thus operate in a parallel way. It is also based on an

application of constitutive equations describing stress, temperature, grain size and stacking

fault energy dependence of creep-rates due to individual mechanisms. Figure 2.2 shows the

deformation map for copper where several regimes are illustrated as a function of temperature

and grain size. The dislocation creep regime is indicated as power-law creep.

2.4 Methods for the determination of Activation Energy of Creep

Based on Eq. 2.1 the apparent activation energy for creep, , is defined as: aQ

2.4) ( ) sσ/G,a /1

)(ln⎥⎦

⎤⎢⎣

⎡∂∂

⋅−=T

RQ ε&

where G/σ and “s” indicate that the data should represent constant normalized stress and

constant structure conditions. In practice, the most frequently used method consists in

performing several creep tests at constant normalized-stress and different temperatures and

is obtained from the slope of the line fitting the steady-state creep-rates vs. the inverse of

temperature (1/T). Nix et al. [

aQ

34] have compiled the activation energies of creep available in

the literature for a large class of metals and compared them with the corresponding activation

energies of lattice self-diffusion, , Figure 2.4. They have shown that the two quantities

are essentially equal, which supports the idea that the mechanism of five-power-law creep

(with stress exponent ≈ 5) is strongly related to diffusion of vacancies.

SDQ

n


Figure 2.4. Activation energy ( ) and volume (SDQ LVΔ ) for lattice self-diffusion versus

activation energy ( ) and volume (aQ aVΔ ) for creep for various metals (from [34]).

Sherby et al. [35], using the same classical method, have obtained the activation energy

of pure aluminum over a large range of temperatures, from about m3.0 T⋅ to . Figure

2.5 a) illustrates the data obtained by Sherby et al. on a strain-rate versus modulus-

compensated stress plot allowing a direct determination of the activation energy for creep.

Their results confirm that above

m9.0 T⋅

m6.0 T⋅ , is comparable to that of lattice self-diffusion, but

below it decreases below . Luthy et al. [

aQ

m6.0 T⋅ SDQ 36] obtained similar results for aluminum

performing torsion creep tests over a range of temperatures from about to mT⋅ 029.0 m93. T⋅ .

Figure 2.5 b) shows another example for calculation of on a strain-rate versus inverse

temperature plot for silver [

aQ

37]. Above m46 T.0 ⋅ the activation energy for silver is about

195 kJ/mol. Below this temperature it undergoes a transition and it is somewhat lower. In the

temperature range from to m 63.046.0 T⋅ mT⋅ the activation energy is in close agreement with

the activation energy for lattice diffusion of silver of 190 kJ/mol. The parallel lines in this

temperature range indicate that the activation energy is not stress dependent. Below m46. T0 ⋅ ,

the lower value of activation energy is probably related with dislocation core diffusion [37]. A

fair amount of experimental evidence on different materials indicates that the effect of the


stress on the values of is small at high temperatures [aQ 38,39,40]. However, in the low

temperature regime the activation energy can be stress dependent as indicated for example in

[40].

An advantage of the classical method is related to the broad range of temperatures and

stresses for which the method can be applied and consecutively to the possibility to determine

an eventual stress or temperature dependence of the activation energy.

Figure 2.5. (a) Steady-state strain-rate versus modulus-compensated stress for aluminium at

various temperatures [35]. (b) Steady-state strain-rate versus inverse temperature for silver at

three different modulus-compensated stress [37].

The definition of activation energy (Eq. 2.4) indicates that only tests performed at

constant normalized stress, )(/ TGσ and characterized by identical structures can be used for

evaluation. This requirement seems to be partly satisfied already by tests performed at

constant )(/ TGσ , when the average subgrain sizes and average distance between free

dislocations were found to obey a power law relationship with )(/ TGσ [41]. Subgrain

misorientations, however, are not constant and slightly evolve with strain [42]. The

importance of subgrain boundaries in creep can be related to their role played in the recovery

of free dislocations, controlling by this the creep-rate [43]. This indicates that samples at

different temperatures and strains have slightly different structures and suggests that an


evaluation method based on the same specimen is more adequate [44-46]. Since temperature

appears in the exponent of Eq. 2.1 a small temperature change can have a large effect on the

strain-rate. Therefore changing the temperature by a small amount might

significantly change the creep-rate

12 TTT −=Δ

1ε& to 2ε& . If the structure does not change much over the

TΔ range the creep-rates and temperatures can be related as [44-46]:

2.5) ⎟⎟⎠

⎞⎜⎜⎝

⎛−⋅=⎟⎟

⎠

⎞⎜⎜⎝

⎛⋅

12

1 11lnTT

Rεε&

&

2aQ .

The procedure can be repeated several times for different temperature jumps and the

activation energy can be obtained from the slope of the line fitted to the logarithm of strain-

rate ratios as a function of the difference of the inverse temperatures (Eq. 2.5). Its application

is illustrated in Figure 2.6. values evaluated according to this method for Al and Ni can be

found in Refs. [

aQ

38,47].

Figure 2.6. Determination of the apparent activation energy by the “differential” method [48].


The activation energies obtained by Refs. [35] and [36] by the classical method for

aluminum are practically the same in the high temperature range (above ) of about

138 kJ/mol and 135 kJ/mol, respectively. Evaluations according to the “temperature jump” or

“differential” method yielded = 147 kJ/mol for aluminum in the same temperature range

[

m6.0 T⋅

aQ

47], which is higher by about 7% compared to results of the classical procedure.

The disadvantage of the differential method for the activation energy determination is

the unavoidable inertia of the furnace in which the temperature must be quickly increased or

decreased specially at high temperatures. The method is applicable at relatively high creep-

rates only, therefore, it has found relatively little use.

Another method for evaluation of relies in the experimentally proven fact that at

homologous temperatures higher than about 0.5 the creep strain is a single valued function of

the temperature-compensated time , i.e. that

aQ

Θ

2.6) constant),( =Θ= σε f .

where is defined by the equation Θ

2.7) dtTR

Qt

∫ ⎥⎦⎤

⎢⎣⎡

⋅−=Θ

0

aexp .

Differentiating Eqs. 2.6 and 2.7 with respect to time and combining them we obtain

2.8) constant,)(exp a ==Θ=⎥⎦

⎤⎢⎣

⎡⋅

⋅ σε ZFTR

Q& .

where Z is the Zener-Hollomon parameter [49] or the temperature-compensated creep-rate

[48]. In this method several creep curves at various temperatures are obtained (over not too

broad range) at a constant stress (Figure 2.7 a). A value of can be determined which

brings together the curves into a single one when plotting the creep strain,

aQ

ε , against the

temperature compensated time, , as shown in Figure 2.7 b). The value of obtained with

this method by Dorn [

Θ aQ

mT⋅48] for aluminum in the temperature range from to 45.0 m57.0 T⋅ is

142 kJ/mol. It is in agreement with values from the last two methods described. A second

variant of this method assumes a direct association between structure and creep strain in the


form )(exp a εε gTR

QZ =⎥⎦

⎤⎢⎣

⎡⋅

⋅= & ,[48]. Experimental data from Figure 2.7 a) plotted according

to this relation show very good correlation for the above value of Q , Figure 2.7 c). a

Figure 2.7. Determination of apparent activation energy of creep by means of temperature

compensated time Θ for aluminum. (a) Creep curves at various temperatures for a given

stress of 21 MPa. (b) Dependence on creep strain on temperature compensated time. (c)

Variation of Z parameter with creep strain [48].

Based on the concept that the effective stress (σ*) for dislocation motion is the

difference between the applied stress σ and the internal stress σi created by neighboring

dislocations and other obstacles, Ahlquist et al. [50] have introduced two other definitions of

the activation energy and at constant effective and internal stress, respectively: *Q iQ

2.9) *)/1(

ln*

σ

ε⎥⎦

⎤⎢⎣

⎡∂∂

−=T

RQ s&

.


2.10) i

TRQ s

iσ

ε⎥⎦

⎤⎢⎣

⎡∂∂

−=)/1(

ln &.

They suggested that the former activation energy characterizes dislocation glide,

whereas the latter the recovery process.

Later Dobeš and Milička [51] have suggested the determination of the activation energy

at the same structural state, i.e. at an identical level of the internal stress, and at the same

applied stress:

2.11) σσ

ε

,)/1(ln

iT

RQ si ⎥

⎦

⎤⎢⎣

⎡∂∂

−=∗ &.

They measure the internal stress at a temperature and applied stress 1T 1σ by the dip

test technique [52,53]. Then a stress change experiment is performed at a temperature and

applied stress

2T

2σ to which the same internal stress value corresponds. They argued (using

experimental data for the aluminum alloy Al–13.7 wt.% Zn) that the extrapolated value of

to zero applied stress becomes identical to the apparent activation energy of creep obtained

from classical tests. The constant structural stress state condition in this definition resembles

the constant structure condition of Eq.

∗iQ

2.4.

It is the aim of the first part of this thesis to demonstrate the adequacy of in situ

tomography for the evaluation of activation energy of creep based on a single specimen by

monitoring its deformation in situ. The method is exemplified on brass and stainless steel

specimens deformed under different conditions and the tomographic outcome is compared

with results from literature and obtained according to the traditional method.

2.5 Creep damage under multiaxial stress

2.5.1 Experimental techniques for damage evaluation

Several techniques have been used to study creep cavitation on grain boundaries. For

example, density measurements allow for the determination of the total cavity volume [54].


Small angle scattering has also been employed to measure the total cavity volume, and under

certain favourable conditions the cavity size distribution can be also obtained [7,14,15].

Optical microscopy and scanning electron microscopy (SEM) are widely used tools to study

cavity distribution and cavity shape [8,9,55-58]. Damage characterization by these techniques,

however, has the drawback that the images analyzed may contain artefacts due to sample

preparation [59]. These drawbacks were to some extent overcome by e.g. the combination of

SEM with a two stage creep device [60]. In this technique a small additional increment of

creep strain is applied to the polished specimens to open up small cavities, which may have

been closed by polishing. Needham and Gladman [61] make small cavities visible by ion

beam etching of the polished sections. For understanding the mechanism controlling damage

accumulation, creep tests have been interrupted at various stages and crept specimens were

fractured in a brittle manner at low temperatures [60,62,63,64]. These techniques have been

used to obtain information about the cavities size and growth kinetics. Since the same cavity

could not be followed due to the descructive nature of the techniques an attempt was made to

measure only the largest cavities (the ith largest or the 100 largest cavities) considering that

they have nucleated at zero time [65]. Creep damage under multiaxial stress develops

inhomogenously and a quantitative assessment is even more difficult by conventional

metallographic techniques. Therefore it has been usually done qualitatively and compared

with results from finite element models (FEM) [66-72]. Serial sectioning combined with

metallographic techniques has been used in an attempt to obtain quantitative information on

damage distribution in specimens subjected to multiaxial creep, such us number of creep void

and area fraction of voids [73-76]. These methods, however, do not allow resolving the real

three dimensional complex shape of the cavities, their connectivity and spatial distribution,

inherent three-dimensional (3D) parameters [77].

More recently, Synchrotron X-ray Microtomography (SXRM) has emerged as a new

powerful technique [78-87] that enables a 3D characterization of heterogeneous

microstructures. The high resolution (~0.33 μm) achievable with microtomography turns this

technique into an excellent tool for the characterization of creep damage at the micrometer

scale being well suited for the study of samples subjected to inhomogeneous loading

conditions. Furthermore, sufficiently large volumes can be measured, yielding relevant

quantitative information: (i) the morphology of cavities, (ii) cavity spatial orientation, (iii)

minimum distance between cavities, (iv) the connectivity between cavities can be captured,

whereas voids may appear separated on two-dimensional (2D) projections. Another advantage

of this technique is that no careful specimen preparation is needed. All the information is


saved in a digital file containing the data of a 3D volume. Its extraction, however, is not a

trivial task. It is the aim of the present investigation to asses the damage distribution in

notched hollow cylinder subjected to uniaxial load and inner pressure, to compare the results

obtained from tomography with those of conventional metallographic techniques and to

evaluate the role of different stress parameters on damage distribution within the notch.

2.5.2 Development of creep resistant steels

Creep cavitation concerns all industries where structural materials are subjected to

mechanical loads at high temperatures. For example, in power plants the increase in thermal

efficiency is achieved by increasing the temperature and, to a lesser extent, the pressure steam

entering in the turbine. The increase in temperature requires new concepts in alloy design. In

this section, a summary of the development of creep resistant steels and published literature

on cavitation in different steels is given.

Investigations carried out with different Mo, Cr, Ni, V, CrMo, MnSi, MoMnSi,

CrSiMo, CrNiMo, CrMnV and CrMoV contents brought forth low alloyed steels for high

temperature applications. In the 1950s, MoV steel (0.14%C-0.5%Mo-0.3%V) was developed

with higher creep strength for steam plants. Molybdenum was recognized as an important

element for improving high temperature strength if the content is about 0.5%. The strength of

these steels is due to solution hardening and Mo2C precipitation. However, a drawback of Mo

alloying over about 0.35% is a marked decrease in ductility under creep conditions as well as

graphite precipitation. This was overcome by the addition of Cr in amounts of 1% and 2.25%.

Steel with chemical composition of 0.15%C-0.3-0.5%Mo, 0.13%C-1%Cr-0.5%Mo and

0.10%C-2.25%Cr-1%Mo were developed and are still in use today. Microstructure

investigations revealed M3C, M7C3 and M23C6 precipitation in 0.13%C-1%Cr-0.5%Mo steel

and Mo2C and M23C6 in 0.10%C-2.25%Cr-1%Mo [88]. In the field of turbine manufacture,

steel with approximately 0.25%C-1.25%Cr-1%Mo-0.3%V was developed. Its resonable

strength is due to finely distributed and thermally very stable V4C3 precipitates and Mo2C.

The highest creep strength of this steel is achieved with an upper bainite structure. In the

1980s, two new low-alloyed heat-resistant steels have been developed with higher creep

strength than the aforementioned ones. The creep strength was achieved by the addition of

Nb, N and B. The names and chemical composition are HCM2S (0.06%C-2.25%Cr-2%Mo-


1.6%W-0.25%V-0.05%Nb-0.02%N-0.003%B) and 7CrMoVTiB (0.07%C-2.4&Cr-1.0%Mo-

0.25%V-0.07%Ti-0.01%N-0.0004%B).

Low-alloy ferritic steels are widely used in the industry at temperatures up to 550°C.

Under service conditions, these steels usually operate within a creep regime where failure

occurs at low ductilities owing to the nucleation and growth of grain boundary cavities.

Numerous investigations were performed in 1%Cr-0.5%Mo, 2.25%Cr-1%Mo and 0.5%Cr-

0.5%Mo-0.25%V steels which are still in use today [61,89,90,91].

With further increase of service temperature, ferritic-martensitic steels with higher Cr

content (typically 9-12%) were developed operating up to 600-650°C. The X22CrMoV-12-1

steel was developed in the 1950s for thin- and thick-walled components up to 566°C. Its

strength is based on solution hardening and on the precipitation of M23C6 carbides. In the

1970s, a newer steel generation the 9CrMo or P91 (designation of the ASTM specifications)

has been developed at the Oak Ridge National Laboratories (USA) [92] for the manufacture

of pipes and vessels for a fast breeder and found broad application in power stations with

steam temperature up to 600°C. The increase of the creep strength in comparison with the

12%CrMoV steel was obtained by secondary MX precipitation of the type VN and Nb(C,N)

due to the addition of 0.05% Nb, 0.2%V and 0.05% N. More recent developments in Japan,

Nippon Steel Corporation, led to the steel NF616 (9%Cr, 0.5%Mo, 2%W) – also designated

as P92 in the ASTM specifications. A further increase in the rupture strength was obtained by

the addition of 1.8% W, 0.003% B and the reduction of the Mo from 1% to 0.5%. The

addition of boron gives thermally stabile M23(C,B)6 precipitates whereas the higher W content

leads to a higher amount of the Laves phase (Fe,Cr)2(Mo,W). In Europe, similar research

activities of the Cooperation in Science and Technology (COST) Action 501 [93] led to the

development of steel E911, which with a similar amount of Cr (9%), less W (1%) and more

Mo (1%) offers similar rupture strength as P92. The Ni content is also low (0.07%), in the

steels P92 and E911, since due to the low Cr content there is no risk of the occurrence of delta

ferrite. Further ferritic 9-10%Cr steels are under development for steam temperature up to

650°C [16]. Figure 2.8 shows the maximum operating temperature of different ferritic and

austenitic steels for 100,000 h creep rupture strength of 100 MPa. The rupture strengths of

high Cr ferritic-martensitic steels are comparable to those of austenitic stainless steels.

Creep failure of high Cr steels is generally caused by loss of strength from accumulated

microstructural damage such as coarsening of precipitates and critical decrease in dislocation

density due to thermally-activated recovery processes accompanied by growth and


coalescence of creep cavities. Publications on cavitation in 9-12% Cr steels are, however, less

frequent [66,74,75,76,94,95].

Austenitic steels were developed for chemical plant equipments and used in various

corrosion and oxidation environments. Austenitic steels exhibit not only good corrosion

resistance but also very high creep rupture strength and therefore they have found applications

for pressure vessels of fast breeder reactors. They can be used at higher temperatures than

ferritic steels, up to approximately 750°C, since the diffusion coefficient in the face-centered-

cubic (austenitic) lattice is about two orders lower than in the body centered (ferritic). Many

but not all austenitic steels develop intergranular cavities depending on the heat treatment and

the impurity content, see for example ref. [60] in 304 austenitic steels at 50% of the melting

temperature. It was also reported that grain boundary cavitation at carbides is one of the most

important factors influencing the degradation of austenitic stainless steels at high temperature

[96,97].

Figure 2.8. Maximum operating temperature of the currently used and the newly developed

power station steels for 100,000 h average rupture strength of 100 MPa [98].


2.5.3 Diffusion models of cavity growth

Due to the economical importance of creep at high temperatures much attention has

been given to the mechanisms of intergranular cavities growth. However, as several

mechanisms can be responsible for creep, as discussed in section 2.3, a practically large

number of cases results which cannot be analyzed deterministically. Therefore, certain

idealizations have to be made giving rise to a certain number of models of nucleation and

growth of intergranular cavities under creep conditions. Cavity growth mechanisms are

usually grouped into four categories: plasticity-controlled cavity growth [5,99], diffusion-

controlled cavity growth [3,4,10,33], constrained cavity growth [11,100,101,102] and

coupling of diffusion and plasticity (power-law creep) controlled cavity growth [6,11,99,103].

A survey of these models can be found in [11,12,13,104,105]. At high stresses and relatively

low temperatures, cavities grow by power-law creep, whereas at low stresses and relatively

high temperatures, growth is determined by diffusional flow.

The following sections focus on the diffusional cavity growth models during creep in

polycrystalline metals and alloys. Since, theoretically, the cavity size at nucleation range from

2-5 nm to about 100 nm [13,106] and plasticity-controlled growth models are applied at high

stresses, they will be not discussed here (this subject is described in, e.g. [12,13]). Cavity

growth by diffusion is discussed starting from the diffusion-controlled cavity growth model

proposed by Hull and Rimmer [3], followed by refinements of this model. This will be

followed by the constrained cavity growth model initially proposed by Dyson [100].

Cavity growth at grain boundaries at elevated temperatures has been suggested to

involve vacancy diffusion. According to this mechanism, cavity growth rate is determined by

the gradient of chemical potential of vacancies, f∇ , in the plane of the grain boundary.

Cavities grow due to migration of vacancies under the influence of this gradient. Cavity

growth rate is also influenced by the shape of voids and the diffusion process. During the

growth of a cavity, atoms are transported from the surface of the cavity to the adjoining grain

boundary where they are deposited. The growth rate is thus expected to be controlled by the

slower of the two mechanisms, namely surface diffusion or grain boundary diffusion.

Therefore, it is necessary to distinguish between diffusive growth mechanisms controlled by

grain boundary diffusion or controlled by surface diffusion.


Grain boundary diffusion controlled growth 2.5.3.1

In the case when grain boundary diffusion is slower than surface diffusion, the growth

process is governed by the first mechanism. The voids retain their spherical cap shape since

surface diffusion rapidly redistributes the matter within it (equilibrium growth). Hull and

Rimmer [3] first proposed a model based on diffusion-controlled cavity growth. The

equations that Hull and Rimmer and subsequently others [4,33] derived for diffusion-

controlled cavity growth are similar. The basic form of their result can be expressed as:

2.12) aTk

aD

dtda

⋅⋅⋅⋅

⎟⎠⎞

⎜⎝⎛ ⋅

−⋅Ω⋅⋅≅

s

mgb

2

2

λ

γσδ.

where δ is the grain boundary width, Ω is the atomic volume, a is the cavity radius, σ is

the remotely applied stress normal to the grain boundary, is the Boltzmann constant, k T is

the absolute temperature, is the grain boundary diffusivity, gbD sλ is the mean separation

between cavities, and mγ is the surface tension of the metal. These parameters are described

in Figure 2.9 which shows the basic geometry for diffusive cavity growth.

Figure 2.9. Cavity growth by diffusion along cavity surface and through the grain boundaries

due to stress gradient.


By integrating Eq. 2.12 between the critical radius (below which sintering occurs) and

the coalescence condition ( 2/sλ=a ), a linearly inverse stress dependence for the rupture

time σ/1r ∝t is obtained.

Later, improvements were made by [4,13,107,108,109] including the influence of

diffusion lengths (the entire grain boundary is a source of vacancies), stress redistribution (the

integration of the stress over the entire boundary should equal the applied stress), cavity

geometry (cavities are not perfectly spherical) and the “jacking” effect (atoms deposited on

the boundaries cause displacement of the grains). Riedel [13] proposed an equation for the

growth rate of unconstrained widely-spaced cavities as:

2.13) ( )

2s

0gb

)24.4/ln(22.1 aaTkD

dtda

⋅⋅⋅⋅

−⋅Ω⋅⋅=

λσσδ

.

where 0σ is the sintering stress. The same inverse stress dependence of the rupture time is

obtained by integrating Eq. 2.13. Despite these improvements, the basic description suggested

by Hull and Rimmer is largely representative of unconstrained cavity growth. An important

point is the predicted inverse stress and activation energy (of grain boundary diffusion)

dependence of cavity growth. When cavities are subjected to a triaxial stress field with

3,21 , σσσ being the principal stresses, the stress used in Eq. 2.13 should be 1σ (maximum

principal stress), the other components 32 ,σσ having less influence on void growth by this

mechanism [11].

The prediction of the stress dependence based on the proposed equations has been

frequently tested. In polycrystals, it has been found that the stress dependency is often

stronger than the predicted one, and is compatible with the Monkman-Grant relationship, i.e.

. Some examples are the studies of Pavinich and Raj [nt σε /1/1r ∝∝ & 110] on copper

polycrystals and of Raj [111] on copper bicrstals. In polycrystals, stronger stress dependence

and ruptures times much longer than the predicted ones were observed. In bicrstals, however,

the rupture time was inversely proportional to the stress, consistent with the diffusion

controlled cavity growth model. In α-brass, Svensson and Dunlop [112] found a linear

dependence of cavity growth with stress, however, the fracture time was consistent with the

Monkamn-Grant and continuous nucleation was observed. Cho et al. [113] and Needham and

Gladman [90] have measured the rupture times and cavity growth rates and found consistency

with the cavity diffusion growth model of Hull and Rimmer if allowance for continuous


nucleation was made. These ambiguous results have led to continuing investigations

regarding the possible reasons of such discrepancies.

Surface diffusion controlled growth 2.5.3.2

When surface diffusion is slow compared with diffusion in the grain boundary, the

cavity ceases to grow as a spherical cap. Matter at the tip of the cavity is removed at a faster

rate and it becomes flatter and more crack-like until the curvature difference between the

poles and the equator (tip of the cavity) is sufficient to drive a surface flux which matches the

boundary diffusion in the grain boundary. The growth of the cavity is then controlled by

surface diffusion. This is also known as non-equilibrium cavity growth. This problem was

first studied by Chuang and Rice [114] and Chuang et al. [10]. They determine the three-

power stress-relationship for the cavity growth at low stresses as:

2.14) 32

m

s

2σ

γδ⋅⋅⋅Ω⋅⋅

≅Tk

Ddtda

where is the surface diffusion coefficient, and the other terms have their usual meaning.

The result is an increasingly crack-like cavity whose growth rate depends on the third power

of the stress. At higher stresses, the growth rate varies as [

sD

2/3σ 13].

Nieh and Nix [115] measured the activation energy for cavity growth in copper

(assuming unconstrained growth) in which vapour bubbles were implanted on grain

boundaries before creep tests in order to bypass the nucleation stage. They also assumed that

continuous nucleation was not important during creep. Their measurements are inconclusive

as to whether it better matches versus . However, the stress and temperature

dependences of the measured lifetimes are in good agreement with an inverse cubic law

( ). Later, Stanzl et al. [

sD gbD

3r /1 σ∝t 116] confirmed the observations of Nieh and Nix on the

same material by performing tension and torsion creep tests. They also found that rupture

lifetime was determined by the maximum principal stress only, but not by the von Mises

equivalent stress.

A problem associated with the surface-diffusion controlled growth model (even when

the inverse cubic law fits the data for copper) is that it is not clear in the experiments why


surface diffusion should be much slower than grain boundary diffusion. For most of the

materials, is larger than [sD gbD 13].

Constrained diffusional cavity growth 2.5.3.3

For the case of grain boundary diffusion or surface diffusion controlled growth it was

assumed that the cavities are uniformly distributed on all boundary facets normal to the

external stress direction. However, over a substantial fraction of the creep lifetime, cavitation

is confined to relatively isolated grain boundary facets which are surrounded by undamaged

material, as illustrated in Figure 2.10. According to experiments the tendency of a grain

boundary to cavitation depends on its orientation to the stress axis and on the crystallographic

orientation of the adjacent grains (i.e. high or low angle grain boundary). If a cavity grows by

diffusion, the material surrounding the cavitating facet should deform also in order to

accommodate the excess of volume of the cavity. Thus, the cavity growth rate may be

controlled by the deformation rate of the surrounding matrix. This was called by Dyson

[100,117] “constrained cavity growth”. If the surrounding material were rigid, the cavity

growth would come to a standstill. On the other hand, if the surrounding material is relatively

soft, the accommodation process occurs readily and cavity growth rate approximate to that of

diffusion-controlled (unconstrained limit). The constraint can be interpreted as the

surrounding material exerts a back stress on the cavitating grain boundary facet. The resulting

stress on the grain boundary, bσ , adjusts itself such that the rate of volume increase by

diffusive cavitation is compatible with the deformation rate of the surrounding material. Here

two extreme cases can be distinguished: (i) the unconstrained limit, in which the cavitated

facet is subjected to the whole applied stress, i.e. , and (ii) the constrained limit, in

which the stress on the boundary is reduced to the sintering stress.

∞=σσ b

Figure 2.10. Uniformly distributed cavitation (a) and heterogeneously distributed cavitation

(b) at transverse grain boundary facets.


Rice [101] developed a quantitative model based on the idea of constrained cavity

growth, which was further improved by other authors (e.g. [13]). The model was idealized by

cavitating boundary facet described as a penny-shaped crack embedded in a creeping matrix

described by Norton’s power law (Figure 2.11). An axisymmetric loading was considered

with being the normal applied stress, and the transverse stress. ∞1σ

∞Tσ

Under the conditions specified above, the cavity growth rate can be described by the

relationship derived by Riedel [13]:

2.15)

⎟⎟⎠

⎞⎜⎜⎝

⎛

⋅⋅⋅

+⋅⋅Ω⋅⋅⋅

⋅

−−=

∞

∞

dq

DTkqah

dtda

2se

e

gb

2

0

'2

)()(

)1(

λεσ

δωψ

σωσ

&

where d is the facet size, is the cavity area fraction on grain boundary facets,

is an abbreviation,

2s )/2( λω a=

2/12 )/31(' nq +=π ∞∞∞ −= T1e σσσ is the von Mises equivalent stress,

is the equivalent strain-rate, and the functions ∞eε& )(ψh and )(ωq account for the cavity

shape and the cavity area fraction, respectively. The other terms have their usual meaning.

Figure 2.11. Constrained cavity growth model on a grain boundary facet.

Needham and co-workers [73,118] have carried out an extensive investigation on low

alloyed ferritic steels (1Cr-½Mo and 2¼Cr-1Mo). In their study, the growth rate of the ith

largest cavity (with i≅1000) has been quantified for both steels. The experimental cavity


growth rates were then compared with the theoretical predictions. The comparison is shown in

Figure 2.12. For the predictions, the sintered stress, 0σ , was neglected and the expression

2/)()( ωψ qh ⋅ was approximated by 1 as a representative average value. The measured

strain-rates and the values for grain size (d=18µm) and for the cavity spacing ( m2.3s μλ = for

the 1Cr-½Mo and 4.5µm for the 2¼Cr-1Mo) were also introduced in Eq. 2.15. The cavity

diameter chosen was m12 μ=a and the material parameters were taken from tabulations

[119]. As it is shown in Figure 2.12, the unconstrained diffusion model overestimates the

growth rate considerably and leads to incorrect stress dependence. However, agreement of the

constrained cavity growth model with the data is very good for the 2¼Cr-1Mo and fair for the

1Cr-½Mo. It should be pointed out, however, that a and sλ vary during the test substantially

whereas the calculations were done using average values. The agreement of the constrained

growth model with Needham’s data suggests that diffusive cavity growth is indeed

constrained under the condition of these experiments.

Cane [62] arrived to a similar conclusion by analysing cavitation in 2¼Cr-1Mo steel at

565°C. The material was heat treated to produce a coarse-grained bainitic structure. The

thermal treatment produced a prior austenite grain size of d=150 µm and fine sulphides on

grain boundaries which act as nucleation sites for cavities. Cavity spacing and cavity radii

were m5.4s μλ = and m8.1 μ=a , respectively. With these parameters, the constrained

growth model agrees with the measurements within a factor of 2.3, while the unconstrained

model strongly over predicts the cavity growth rate at low stresses.

In a later work, Cane [120] investigated notched specimens of the same steel. For the

notch geometry considered , and according with Eq. 2/ e1 =∞∞ σσ 2.15 the growth rate is

expected to be twice larger than in the uniaxial tension case for the same equivalent von

Mises stress. Although few measurements were presented, a reasonable close agreement was

obtained, being the growth rate under multiaxial stress state a factor of 1.5 larger than the data

for uniaxial tension.


Figure 2.12. Cavity growth rate for two ferritic steels (1Cr-½Mo and 2¼Cr-1Mo). Predictions

from unconstrained and constrained models are depicted (adapted from [13]).

The time to cavity coalescence on the cavitating boundary facet can be obtained by

integrating Eq. 2.15 between (cavity radius at time t=0, considering nucleation to occur at

the beginning of the test) and

0a

2/sλ=a (cavity coalescence) as:

2.16) d

nhD

Tkht⋅⋅⋅+⋅

+⋅⋅⋅Ω⋅⋅⋅

= ∞∞∞∞ )/()/31()(4.0)(006.0e1e

s2/1

1gb

3s

c σσελψ

σδλψ

&

where n is the steady state stress exponent.

The time to coalescence, , is composed of two terms, the first representing the time to

coalescence in the absence of constraint (i.e. only diffusion controls growth) while the second

term represents the constraint. At small strain-rates, the second term leads to a very long time

to coalescence. In the uniaxial case, the constrained term predicts a much stronger stress

dependence, . It should be emphasised that failure is not expected by coalescence of

ct

nt σ/1c ∝


cavities on isolated facets, and some additional time may be required to join facet-size

microcracks. Further nucleation may be necessary to produce the rupture (e.g. local nucleation

ahead the crack or nucleation on non-cavitated facets). Therefore, continuous nucleation

should be considered in addition to the constrained case. If this is done, the stress dependence

of the rupture time takes the form [5/)23(r /1 +∝ nt σ 13]. The additional effect of continuous

nucleation is illustrated in Figure 2.13 for the rupture life of 2¼Cr-1Mo steel measured by

Cane [120]. For unconstrained growth the difference between predicted values and measured

values of times to rupture is significant, while it is nearly negligible for constrained growth.

No adjustable parameters were used in this study. Figure 2.13 shows an excellent agreement

between and for constrained cavity growth indicating that the time to coalescence, , is

most of the specimen lifetime, in this case.

ct rt ct

rt

Figure 2.13. Time to rupture as a function of stress for constrained and unconstrained cavity

growth with instantaneous and continuous nucleation for 2¼Cr-1Mo steel measured by Cane

(adapted from [13]).

2.5.4 The cavity size distribution function

As it was previously mentioned, cavities nucleate continuously during creep of metals

and engineering alloys and this fact needs to be considered in the lifetime calculation.


Experimental results on cavity nucleation are scarce due to difficulties related to the small

cavity size. The cavity size distribution function provides, however, a link between

experimental data and theories on cavity nucleation and growth. The cavity distribution

function is denoted by , where ),( taN daN ⋅ is the number of cavities per unit grain

boundary area having a radii between a da+ . In this formulation, only those boundary facets

which effectively contribute to the rupture process are considered in the cavity distribution

function. In the case of failure by diffusive cavity growth, the boundary facets that should be

considered are those which have orientations between 60° and 90° with respect to the

principal stress axis, since these boundaries cavitate preferentially. Nevertheless, this freedom

in definition does not affect the results.

The cavities are considered to fit in a certain size class which is determined by its

radius, , but they can pass from one size class to the next one,a daa + , by growing. The

coalescence event during the late stages of the ruptured process is ignored. Riedel [13]

proposed a cavity distribution function by imposing a continuity condition in the size space,

and assuming a power-law functions for the cavity growth rate, , and the

nucleation rate, , as:

dtda /=a&

*J

2.17) . αβ −− ⋅⋅= taAa 1&

2.18) γtAJ ⋅= 2* .

where , , 1A 2A α , β and γ might depend on stress, on temperature, on material parameters

as well as on the actual microstructure, but not on time nor on the cavity size. The final result

for the distribution function proposed by Riedel [13] is:

2.19) )1/()(

11

1

1

2

111),(

αγα

α

βγαβ

βα

−+

−

++

⎟⎟⎠

⎞⎜⎜⎝

⎛+−

−=tA

ataAAtaN

The solution for different values of α , β and γ is shown in Figure 2.14 for

. If 121 == AA 1<α , the distribution function must be cut off at a maximum . If a 1>α , the

distribution extends to infinite , because the first cavities nucleated grow infinitely fast for a

value of

a

1>α . For the case of 1=α , Eq. 2.19 can be rewritten as:


2.20) ⎟⎟⎠

⎞⎜⎜⎝

⎛++−

=+

+

1

11

1

2

)1()1(exp),(

Aata

AAtaN

βγ β

γβ

It is important to note in this case, that the maximum of the distribution function, as

well as the average cavity size, remain at a fixed value of . a

Figure 2.14. Evolution of the cavity size distribution function (adapted from [13]).

2.5.5 Prediction of creep rupture time (phenomenological approach)

Creep rupture by cavity growth and coalescence is an important failure mechanism for

high temperature components. Cavity growth is a very complex phenomenon involving

kinetic and mechanical processes which are manifold (e.g. lattice, grain boundary and surface

diffusion, dislocation creep, grain boundary sliding). Furthermore, the cavity size distribution

is a function of the time and varies with the local microstructure and with the heterogeneous

and continuous nucleation of new cavities during creep. Also, the models which describe the

intergranular cavity growth under creep conditions are based on idealization of the real

structure. The cavity growth, the assessment of the life time and the failure modes under creep

condition have been extensively studied both theoretically and experimentally, e.g.

[5,10,11,100,109,117,121-124] mostly under uniaxial stress conditions. However, in most of

the cases the engineering components are subjected to inhomogeneous loading conditions of

temperature and stress. To be able to predict the failure mode and life time of such industrial

components the assessment of damage on specimens deformed under conditions similar to

those in service is necessary [73]. This is done for example by performing creep tests on

hollow cylinder samples, where the stress multiaxiality is imposed by an internal pressure, a


longitudinal force and the notched sample geometry. Creep lifetime and failure mode under

multiaxial stress conditions can differ significantly from the rules established for uniaxial

loading. For example Cocks and Ashby [5] have predicted that cavity growth is highly

accelerated by high positive stress triaxiality.

Data obtained from uniaxial stress experiments have led to a good understanding of the

physical processes involved, although they do not always provide sufficient information to

predict failure under multiaxial stress conditions. Based on principles of damage mechanics

[125,126] Hayhurst et al. [127-130] have studied the effects of stress multiaxiality on creep

rupture. For a smooth round bar under uniaxial loading, the rupture lifetime, , at a given

temperature is usually expressed as

ft

2.21) . χσ −⋅= Mt f

where σ is the stress an M and χ are temperature dependent parameters. Hayhurst [130]

has shown that Eq. 2.21 does not correctly predict the creep rupture properties of notched

rounded bars and generalized it by including the stress invariants, and [1J 2J 130]:

2.22) . χγβσα −⋅+⋅+⋅⋅= )( 211f JJMt

where 1σ is the maximum principal stress with 321 σσσ >> , H1 3 σ⋅=J ( Hσ is the

hydrostatic stress) and is equal to 2J eσ , the equivalent von Mises stress. The coefficients α ,

β and γ are weighing factors describing the relative contribution of the different stress

parameters to rupture life ( 1=++ γβα ). The physical basis for including various multiaxial

stress terms in Eq. 2.22 is based on the following facts. First, intergranular fracture usually

occurs by diffusive growth of intergranular cavities (driven by the tensile stresses) at

boundaries perpendicular to the maximum principal stress, 1σ . Second, cavity growth at high

temperature can be also driven by the tensile hydrostatic stress component, thus H1 3 σ⋅=J .

Third, high stress concentration produced by inhomogeneous plastic deformation such us

grain boundary sliding or slip band formation are required for the nucleation of cavities.

These deformation processes are driven by shear stresses, therefore, some form of the shear

stress should enter in the equation. Therefore, the effective von Mises stress was also included

in Eq. 2.22. Reviews of multiaxial creep rupture data for several metals suggest that the


maximum principal stress 1σ , and the von Mises equivalent stress eσ are more important than

the hydrostatic stress Hσ in determining the creep lifetime [130]. Then, Eq. 2.22 can be

rewritten as

2.23) . χσασα −⋅−+⋅⋅ ))1(( e1=ft M

where α is a single parameter that describes relative importance of 1σ in the creep rupture

time. The value of α varies for different materials and fracture modes [131]. Assuming that

cavitation is concentrated on grain boundaries (GB) perpendicular to the maximum principal

stress [60,62,116,131-134], Nix et al. [135] have developed a model based on the “principal

facet stress”, which predicts good results in cases when the grain boundary sliding (GBS)

mechanism is active. They argue that GBS relieves the shear stress along the grain boundaries

and the stress causing cavitation is not the nominal applied stress but the concentrated normal

stress acting on the transverse or nearly transverse grain boundary facets. Based on three-

dimensional numerical analysis performed by Anderson and Rice [136], who have used the

Wigner-Seitz cell of an fcc lattice as the grain for modeling, Nix et al. define the average

principal facet stress, , as Fσ

2. 24) )(62.024 321.2fσ = σσσ +−⋅ .

with 321 σσσ >> being the principal stresses that exist in macroscopic sense. It should be

emphasized that in this approach the principal stress is expected to be valid after grain

boundary sliding has caused redistribution of the stresses but before cavitation started.

The hydrostatic stress (or mean normal stress) is known to influence cavity growth [5]

under power law creep. However, Sakane et al. [137] have confirmed its contribution to void

nucleation, too, during experiments with equi-triaxial stress state, i.e. zero von Mises

component. Watanabe et al. [66] and Li et al. [75] also reported that in 9Cr-1Mo-V-Nb steel

welded joints the distribution of the stress triaxiality factor coincided better with the creep

damage distribution rather than the equivalent creep strain. They argued that the local

concentration of both triaxiality factor and equivalent creep strain accelerate void formation

and growth.

More recently, various experiments have been done to test materials under multiaxial

loading conditions. Niu et al. [69,70] analysed creep rupture using tubular and notched


specimens of austenitic steel SUS310S with high ductility, in tension, torsion and combined

tension–torsion stress states at 700°C. They have found that the maximum principal stress

determines the multiaxial creep rupture life of the steel. They further suggest that the fracture

mode is related to the magnitude of Hσ with respect to 1σ and eσ , because is supposed that

Hσ promotes the growth of voids. Sakane et al. [137,138] analysed two- and three-cruciform

specimens subjected to axial forces. They conclude that the von Mises equivalent stress and

the equivalent stress based on crack opening displacement were a suitable parameter to assess

the biaxial creep damage in 304 type stainless steel in which the fracture mode was mostly

transgranular. Hsiao et al. [67] have compared three different stress parameters, the von Mises

effective stress, the maximum principal stress and the principal facet stress to correlate the

local multaxial stress with the local creep damage distribution and failure lifetime on

weldments of 316 steel. They showed that the principal facet stress parameter gave the best

prediction of the creep damage distribution in the weldments.

Based on these experiments, it is suggested that the von Mises stress, eσ , predicts better

the rupture time and creep damage when the fracture mode is transgranular, which is usually

found in short time creep. It has been also reported that the von Mises equivalent stress will

determine the rupture life for metals which show virtually no cracking [139]. The maximum

principal stress, 1σ , on the other hand, will determine the rupture life time of metals pre-

cavitated or continuously cracking, in which many voids have nucleated in the early stages of

creep [70,116]. Furthermore, for metals and alloys which exhibit grain boundary sliding and

fail by cavitation at grain boundaries perpendicular to the maximum principal stress, the

principal facet stress, Fσ , will correctly predict creep rupture life under various stress states.

However, analysis of creep damage under a multiaxial stress state has led to ambiguous

results of the parameters defining creep lifetime and damage distribution. The choice of the

best damage indicator tended to be material, stress and temperature dependent.

The stress parameters usually used for the assessment of rupture life are, however,

averages of the local values in the sample. Under multiaxial loading the damage develops

inhomogeneously indicating that local parameters should be considered for evaluation of the

damage distribution. As mentioned in section 2.5.1, tomography has evolved as a powerful

tool for such evaluations. It is the aim of the second part of this thesis to assess the damage in

a notched hollow cylinder of E911 steel, which was subjected to about 26,000 h of creep of

similar loads as in service. The 3D shape and size distribution of single cavities will be


described. A comparison of local damage distributions with the distribution of selected stress

parameters, knowing to influence damage, will be performed. The tomographic results will

also be compared with those of conventional metallography.

III. Experimental Details 37

CHAPTER III

EXPERIMENTAL DETAILS

3.1 Uniaxial in situ experiments


For in situ tomographic investigations a common easily machinable brass alloy and a

commercial stainless steel (AISI 440B) were chosen as model materials. Their chemical

compositions are given in Table 3.1.

The brass alloy was fabricated by indirect extrusion (1023 K -750°C- billet temperature,

19:1 extrusion ratio, 285 mm/s extrusion rate) at the Extrusion Research and Development

Center of Technische Universität Berlin (TU Berlin) [140]. In order to promote the damage

during creep the sample was pre-strained in tension (4.5% in total) with the specimen axis

parallel to the extrusion direction [141].

Brass (wt.%)

Cu Zn Pb

58 40 2

Steel AISI 440B (wt.%)

C Si Mn P S Cr Mo V

0.85 - 0.95 ≤ 1.00 ≤ 1.00 ≤ 0.04 ≤ 0.015 17.0 - 19.0 0.90 - 1.30 0.07 - 0.12

Steel grade E911 (wt.%)

C Si Mn Cr Mo Ni W V Nb N B

0.11 0.2 1.00 9.1 0.96 0.25 0.98 0.21 0.082 0.091 0.0071

Table 3.1. Chemical composition of the investigated brass alloy, stainless steel AISI 440B

used for activation energy determination and that of E911 steel used for damage evaluation

after 26,000 h of creep.

Specimens for in situ tests were specially developed to meet the requirements of the

tomographic set-up at ID15A of the European Synchrotron Radiation Facility (ESRF). The


geometry of the investigated specimens is presented in Figure 3.1. The sample has a diameter

and gauge length (L) of about 1 mm. Figure 3.1 also shows the position of the illuminated

volume in the steel sample B (see Table 3.2) and its reconstruction by tomography. The

specimens are designed with threads at both ends which are screwed at one end into the load

cell and at the other end into the reaction yoke of the creep machine. The specimens have two

small holes placed 4 mm apart from the center of the sample where thermocouples (T1 and

T2) for temperature measurement are placed.

Figure 3.1. Schematic drawing of the specimen used for in situ investigations. The

illuminated volume in the sample and its reconstruction for the steel sample B by tomography

are also shown.

3.1.2 Creep device for in situ experiments and testing conditions

A small creep machine for in situ experiments was specially developed to fit on the

goniometer table. The machine had a reduced weight to allow for the proper rotation during

tomographic scans. The design of the machine allows a free path for the X-ray beam to reach

the sample during 360° turn around a vertical axis. The loading system is supported by a

ceramic tube made of Macor (Figure 3.2) [142], which has good creep resistance at high


temperature and transparent to high energy X-rays. The lower end of the specimen is screwed

in the reaction yoke of the machine and the upper part is connected to the loading system. The

load is applied to the specimen via two springs (material spring steel 1.1200 Class C) with

proper constant to achieve the required stress condition. When the springs are compressed the

load is transferred to the sample through the load cell, which measures the actual load applied

to the specimen. The high temperature necessary for creep is obtained by resistive heating

coils placed around the ends of the sample. At the position of each heating coil thermocouples

measure the temperature and a proportional-integral-derivative (PID) controller is used to

keep the temperature constant within ±1 K. Asymmetric heating of the sample becomes

possible by setting different temperatures for the two coils, enabling tests with a controlled

temperature gradient along the gauge length. Asymmetric heating produces a temperature

distribution along the stress axis, which is linear in the region of the sample with constant

cross-sectional area. The elongation of the sample is measured with an inductive displacement

transducer attached to the head of the load cell. Recording of data as well as the control of the

machine are made through computer.

Figure 3.2. Creep machine used for in situ measurements.


Experiments with asymmetric heating were performed with brass and AISI 440B

stainless steel. For the brass sample (named as sample A in Table 3.2) the load was supplied

by a spring with small spring constant [87], the stress at the beginning of the experiment being

about 25 MPa and the temperature at the bottom of the sample 673 K. Two experiments were

performed with the AISI 440B stainless steel: the first test at 180 MPa and 985 K at the

bottom of the sample, and a second test at 210 MPa at 1013 K also at the bottom of the

sample. Sample designations (B and C) and corresponding deformation conditions are given

in Table 3.2. In all cases the upper coils were switched off. The asymmetric heating caused a

temperature difference between the two thermocouples (separated by a distance of 8 mm

length) of about 390±2 K (sample A), 624±2 K (sample B), and 645±2 K (sample C).

Sample σ

[MPa]

bT

[K]

TΔ

[K]

T∇

[K/mm]

ssε&

[1/s]

rt

[h]

A (brass) 25 673 ± 1 117 ± 2 37 2.1·10-6 7.3

B (steel) 180 985 ± 1 351 ± 2 99 3.1·10-6 24.7

C (steel) 210 1013 ± 1 372 ± 2 105 1.2·10-5 7.9

Table 3.2. Measurement conditions for in situ experiments. Applied stress σ , temperature at

the bottom of the sample bT , measured temperature difference between calibration points

TΔ , temperature gradient in region with constant cross section T∇ , minimum steady-state

strain-rate ssε& and rupture tim e rt .

3.1.3 Creep tests with nearly constant load or constant stress

A proper selection of the spring constant enables creep tests at nearly constant load or

constant stress. Evidently the elongation of the sample produces a load decrease by the

expansion of the springs, however, this change can be kept below a certain tolerance by

selecting springs with proper constants. The admissible relative load change or tolerance is

defined as follows:

3.1) tolerancex

xxF

FFF ≤−

=−

=Δ0

r0

0

r0 ,


where F0 and Fr are the loads at the beginning and at the end of the creep test and and

the corresponding compression lengths of the springs. The maximum spring constant

fulfilling this criterion is obtained as:

0x rx

3.2) )1( r

0s

00max −⋅⋅

⋅⋅= εelN

toleranceAk σ ,

where is the initial length of the sample, the initial cross section, σ0 the initial stress, Ns

the number of springs and εr the strain to rupture. In our experiments usually a load tolerance

of 5% was selected.

0l 0A

A nearly-constant stress test becomes also possible if the relaxation of the load due to

the elongation of the springs equals the decrease in load due to contraction of the cross

sectional area (assuming the stress constant) of the specimen. The equal stress condition will

be valid at two strains, which can be adequately selected for example at the beginning and at

the end of secondary creep regime. The following condition can be written:

3.3) )()( 21 εσεσ = ,

where 1ε and 2ε are the selected strains. Assuming that the volume of the sample remains

constant during deformation, the initial compression of the spring can be worked out: 0x

3.4) 12

1122 )1()1(00 εε

εεεε

eeeeeelx

−⋅−−⋅−

= ,

with being the initial gauge length. 0l

The spring constant is determined by the selected stress 0σ and is given by:

3.5) ))1(( ii

00s

00

−⋅−⋅⋅⋅

= εεσ

elxeNAk ,

where iε denotes either 1ε or 2ε . Selecting for example the strains 08.01 =ε and 3.02 =ε

where the equal stress condition is imposed it can be shown that the relative stress variation in


the [0, 0.35] strain interval is less than 2%. Figure 3.3 shows the relative stress deviation from

the imposed value of 25 MPa vs. true strain. The stress changes less than 2% during the creep

test if the values of ε1=0.08 and ε2=0.3 are chosen to fulfill the Eq. 3.3.

Figure 3.3. Relative stress deviation during creep test for selected parameters ε1=0.08 and

ε2=0.3, , mm10 =l MPa250 =σ and . 20 mm78.0=A

3.2 Multi-axial creep experiments on E911 steel


conditions

Creep experiments under multi-axial loading conditions were performed at the MPA

Stuttgart using the sample geometry shown in Figure 3.4. Notched hollow cylinder samples

made of E911 steel (chemical composition is given in Table 3.1) were subjected to a uniaxial

load of 8543 N applied along the cylinder axis and an internal pressure of 17.5 MPa. The

temperature was set to 848 K (575°C), which is the usually operating temperature of this

material in a power plant. After a creep time of 26,000 h the test was stopped and a small

cylindrical sample of about 0.6 mm in diameter (area of about 0.28 mm²) and approximately

2 mm in length (this being the minimum width at the notch) was extracted for tomographic

measurements from the middle plane of the notch. Further metallographic investigations were


performed on the same sample at three cross-sections: near the outer, the inner surface and in

the middle of the small cylindrical specimen. A careful preparation with grinding and

polishing was performed. Polishing was done in different steps with graded grain sizes up to 1

µm and a regular change of polishing direction. In addition no etching was applied in order to

prevent the extraction of particles by the reagent. The identification of the cavities was made

by an experienced metallographer using a light-optical microscopy with up to 1000 times

magnification. A representative area of approximately 0.14 mm² was evaluated.

Figure 3.4. Schematic drawing of the notched hollow cylinder specimen creep deformed

under multiaxial loading conditions. For tomographic investigations a small sample from the

middle part of the notch was extracted.

3.3 Synchrotron X-ray microtomography

The experiments in the first part of the present thesis are focusing on in situ monitoring

by microtomography the geometry of creeping specimens subjected to uniaxial load. The

basis of this technique is briefly explained in the next subsection. In situ fast tomography

measurements during creep were performed at beamline ID15A of the ESRF, which was

equipped with a CCD camera having a restricted field of view (FOV) of about 1.2 mm x

1.1 mm. The effective pixel size of the detector was 1.6 µm and had a resolution of about

2.1 µm (the full width at half maximum of the point spread function). The tomograms were

recorded using a high-energy beam (~80 keV) with a large bandwidth (~50 %). The

reconstructed volume is constituted of voxels, each voxel representing a volume of 1.63 µm3.

The acquisition time of a complete tomographic scan was about 3 min. Due to the highly


brilliance beam available at the ID15A, the recorded radiographs had good statistics, which

allowed for good quality reconstructions.

The second part of this thesis focuses on the characterization of damage developed

during multiaxial creep test using X-ray microtomography. A sample extracted from a crept

hollow cylinder was investigated by microtomography at the ID19 beamline of the ESRF

using a monochromatic beam of 51 keV. The beamline was equipped with a CCD camera

having a restricted FOV of about 0.7 mm x 0.7 mm, the effective pixel size of the detector

being 0.33 µm. In this case each voxel represents a volume of 0.33³ µm³. The acquisition time

of a complete tomographic scan was about 2 h. Due to the restricted FOV of the CCD camera

three volumes were measured in order to cover the whole length of the sample of about 2 mm.

3.3.1 X-ray tomography: Principles

X-ray tomography is a non-destructive imagining method in which a cross-sectional

view of an object can be obtained from transmission data collected at many different angles.

The basis of X-ray tomography is X-ray radiography, which represents a “projection” of the

absorption coefficient of the investigated material. The mathematical formulation for

reconstructing an object from multiple projections is based in the work of Radon [143], who

demonstrated the possibility of replicate an object from a set of its projections. X-ray

radiography physics is based on the Lambert-Beer law which relates the ratio of transmitted

(I) to incident (I0) intensity (number of photons) to the integral of the linear absorption

coefficient of the material μ along the path L :

3.6) dxEyxII

L),,(ln

0∫=⎟⎟

⎠

⎞⎜⎜⎝

⎛− μ

μ depends on the material and the X-ray energy, therefore for a polychromatic beam, Eq. 3.6

has to be integrated over the whole energy spectrum.

Because the projections contain superimposed information of a volume in a 2D plane

the estimation of the absorption coefficient distribution of the scanned object needs many

projections to be obtained. For every angular position, θ , a 2D projection image is recorded.

The cross-section of the object (slice) to be reconstructed is called and a parallel ),( yxf


projection of this object at the angle θ is denoted by ),( θtp , where t is the distance from the

projection ray to the center of rotation (Figure 3.5).

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

reconstruction itself. This is the inverse of the Radon transform. In the case of parallel beam

geometry, a condition that can be met at synchrotron sources, the slices of the sample

corresponding to different heights in the sample can be treated independently and it is also

sufficient to record the projections for half turn due to mirror symmetry, i.e.

),(),( θπθ tptp −=+ .

The theory governing the tomographic reconstruction is generally known as the Fourier

slice theorem [144,145]. It states that the Fourier transform of a parallel projection ),( θtp of

an object obtained at an angle ),( yxf θ is identical to a section in the two-dimensional

Fourier transform of taken at the same angle. ),( yxf

3.7) θθω )(),( FP = ,vu

where ),( θωP denotes the Fourier transform of ( ),θtp . This is illustrated in Figure 3.5.

Figure 3.5. Principle of tomography and illustration of the Fourier slice theorem. The object

is represented in the rotated coordinate system by . ),( yxf ),( ts'f


Although the Fourier slice theorem provides a straightforward solution to the

reconstruction problem, it has some inconvenient in actual implementation such us

interpolation in the frequency domain or implementation of the targeted reconstruction [146].

Therefore, alternative implementations of the Fourier slice theorem were explored being the

most popular one the so-called Filtered Back Projection (FBP) algorithm [144,145]. The idea

of the backprojection relies in assigning to each point of the object the average value of all the

projections that pass through that point. The backprojected image is, however, a blurred

version of the original object. An exact mathematical correction of the backprojection

smoothing effect can be performed by an appropriate pre-filtering of the projections leading to

the Filtered Back Projection algorithm. Based on the Fourier slice theorem and symmetry

property given by the parallel sampling geometry it can be written:

3.8) θωωθωπ θθπω ddePyxf yxj∫ ∫

∞

∞−

+=0

)sincos(2),(),(

The inside integral is the inverse Fourier transform of the quantity ωθω ),(P which

represents a Fourier transform of a projection filtered by a function whose frequency domain

is ω (ramp filter). The ramp filter emphasizes high-frequency contents and consequently the

high-frequency noise. Different types of windows can be applied to shape the filter’s

frequency response and hence modify the noise characteristic to the reconstructed images.

Some often used window functions are Hanning or sinc functions [146]. All tomographic

volumes evaluated in this thesis were reconstructed using the FBP algorithm with a

Butterworth filter [144].

3.4 Image processing

3.4.1 Reconstruction and pre-processing

The reconstructions of tomographic measurements were performed with in-house

developed software and their processing was performed using algorithms written in

Interactive Data Language (IDL) and Matlab. For damage evaluation purposes histogram

equalization was applied to each slice in the volume. When necessary, the remaining ring

artifacts in the reconstruction (visible at the center of Figure 3.6a) were removed by an


algorithm based on local threshold and finally a band pass filter was applied to enhance the

contrast of the cavities and make them easier to segment. The band pass filter procedure was

applied, however, only for counting the cavities. A mask situated 50 pixels inside the border

of the sample was then used to avoid picking up artifacts coming from the band pass filtering

(light border at sample surface visible in Figure 3.6c). The sequence of the procedure applied

for one slice is exemplified for the E911 steel sample in Figure 3.6.

Figure 3.6. Sequence of image analysis applied for pore number extraction. a) Original slice

as obtained from reconstruction, b) slice after histogram shift and ring artifacts removal, c)

band pass filtering, d) mask and threshold applied for pore extraction.


3.4.2 Identification of cavities

The extraction of the cavities was performed by identifying voxels as belonging either

to a cavity or to the matrix material based on their gray level. For this, an algorithm based on

local grey level threshold was written in IDL. A cavity is considered as a configuration of

voxels, which share a common face, edge, or corner. Cavities with a size of 2 x 2 x 2 or more

connected voxels were considered. Smaller cavities can result from noise and were neglected.

In order to characterize their shape and orientation, creep cavities were approximated by

ellipsoids with the same volume and moment of inertia [147,148]. A diagonalization of the

momentum of inertia tensor, I, can be performed in order to obtain the eigenvalues, which

allow calculation of the length of the semi-axes, , and the eigenvectors, representing the

spatial orientation of the semi-axes. The semi-axes length can be calculated as [

ja

149]:

3.9) 1,2,3j,2

)2)((5 jj =

⋅

⋅−⋅=

VIITr

a

where V is the cavity volume and is the trace of the momentum of inertia tensor. It is

also assumed that the mass density is equal to unity. A description of the cavities fitted by

their equivalent ellipsoid can be given by the definition of 3D shape descriptors namely

elongation, , and flatness, [

)(ITr

e f 150]. The elongation parameter used in this work, e , has been

slightly modified in order to obtain the ratio between the length of the maximum semi-axis,

, and the average of the two minor semi-axis, and , respectively. Elongation and

flatness are then defined as:

1a 2a 3a

3.10) 32

12aa

ae+⋅

=

3.11) 3

2

aaf =

With these two parameters, the shape of nearly regular cavities can be described and

quantitatively separated in different cavity shapes such us spheres, ellipsoids and rods [83].

However, these shape descriptors cannot provide information about the complexity of the


cavity shape, therefore and additional parameter (complexity factor, CF ) was proposed [83].

The parameter CF indicates the deviation of the cavity shape from its equivalent ellipsoid. It

is related to the exclusion volumes resulting from the intersection of the real cavity with its

equivalent ellipsoid. Figure 3.7 illustrates this definition in a 2D case. The complexity factor

is defined as:

3.12) 21TT

31 , VVVV

VVCF +=

+=

A large CF indicates that the void’s shape is different from that of an ellipsoid, while a

small value means a good similarity. CF varies between zero and about two and offers a

good possibility to distinguish between regular cavities and those affected by coalescence.

Figure 3.7. Complexity factor, CF , definition based on a 2D projection, a) 2D projection of a

real cavity, b) equivalent ellipsoid, d) both.

The orientation of the cavities was characterized by the angle between the major axis of

the equivalent ellipsoid and an arbitrary axis which can be defined conveniently in each

experiment. Usually for analysis of cavities under creep condition the loading axis is chosen

as the reference axis [83]. However for the E911 sample, the reference axis was considered

along the notch radius.


3.4.3 Image correlation for evaluation of tomographic data

To monitor changes in the sample’s shape, material slices lying perpendicular to the

external stress axis were chosen, which due to the axial symmetry of the experiment were

supposed to move during creep along this axis. Slice displacement was evaluated by cross

correlating a selected slice from the initial volume with slices from consecutive

reconstructions.

The classical bi-dimensional cross-correlation method quantifies the degree of fit of two

matrices, and , to a linear model using the correlation coefficient. For image-

processing applications, variation in the brightness of the image can lead to erroneous values

of the correlation coefficient. In order to decrease this effect, the images are normalized by

subtracting the mean and dividing by the standard deviation. The cross-correlation coefficient

is defined as:

nm × mnX mnY

3. 13) [ ][ ]

∑∑∑∑

∑∑−

=

−

=

−

=

−

=

−

=

−

=

−−

−−

−−−=

1

0

1

0

21

0

1

0

2

1

0

1

0

)(1

1)(1

11

1

N

m

N

nmn

N

m

N

nmn

N

mmn

N

nmn

YYN

XXN

YYXXNr

where is the number of pixels in one image, and are the pixel intensities of the

images and

N mnX mnY

X and Y their average intensities, respectively.

The correlation coefficient, r , is a scalar quantity which varies in the interval [-1,1]. A

value of 1+=r indicates a perfect fit to a positive linear model whereas 1−=r is a perfect fit

to a negative linear model. Although conventional correlation can provide information about

similarities between two images, it is inefficient in the presence of horizontal shifts (image not

in phase with each other), which is the common situation in deformed samples due to material

flux towards the center of the sample. Moreover, the standard cross-correlation technique is

sensitive for luminance variations, which requires pre-processing of the images.

An advantage for in plane shift detection with respect to the classical cross-correlation

is provided by the Phase Correlation method, based on the translation property of the Fourier

Transform and expressed by the Fourier-shift theorem [151]. The theorem states that a

translation of a given function in the space domain corresponds to a phase shift in the

frequency domain, which is represented by Eq. 3.14 and 3.15.


3.14) ),(),( 0012 yyxxfyxf −−=

3.15) ),(),( 1)(2

200 vuFevuF yvxuj ⋅= ⋅+⋅⋅⋅⋅− π

where and represent two grey level images, respectively, is the

shift of the function with respect to

( yxf ,1 ) )

)

( yxf ,2

( yxf ,2

),( 00 yx

( )yxf ,1 in the space domain and

denotes the Fourier transform of , = 1, 2.

),( vuFi

),( yxfi i

Then, the cross-power spectrum is defined as:

3.16) )(2*

12

*12 00

),(),(),(),( yvxuje

vuFvuFvuFvuFG ⋅+⋅⋅⋅⋅−=

××

= π

where * indicates the complex conjugate. The inverse Fourier transform (IFT) applied to

Eq. 3.16 gives the Dirac delta function ),( 00 yyxx −−δ centered at expressed in

Eq.

),( 00 yx

3.17.

3.17) ( ) ),(),(),(),(),(

00)(21

*12

*121 00 yyxxeF

vuFvuFvuFvuFF yvxuj −−==⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

×× ⋅+⋅⋅⋅⋅−−− δπ

Then, from the inverse Fourier transform ( 1−F ) of the cross-power spectrum a peak

corresponding to the Dirac delta function, is obtained at the position of the shift between the

images. Other advantages of phase correlation are the possibility to detect “sub-pixel”

changes [152], and the small sensitivity to luminance variations. The most sensitive

correlation indicator was found to be, however, the product between the amplitude of the

inverse cross-power spectrum and the classical cross-correlation coefficient [153]. The

presented algorithm was applied to monitor sample shape changes during creep.

IV. Results 52

CHAPTER IV

RESULTS

4.1 Method for evaluation of apparent activation energy of creep, a Q

The asymmetric heating applied to the samples in the in situ experiments produces a

temperature gradient (Table 3.2) which can be used to determine the temperature dependence

of strain and strain-rate along the stress axis. This dependence can be exploited to evaluate

as it will be explained next. However, the evaluation of the local strain and strain-rate is

only possible if we can correlate or follow the displacement of slices perpendicular to the

external stress during the creep process. This was done using the cross correlation method

explained in section

aQ

3.4.3. The temperature distribution in the specimen can be determined by

calibration measurements combined with finite element (FE) calculations. A description of

the calibration procedure will be given in section 4.1.1. Figure 4.1 shows the reconstruction of

a brass sample after 307 min of creep. The slab, composed of 10 slices, indicates that the local

strain is a function of temperature coupled with the position along the Z axis.

Figure 4.1. Tomographic reconstruction of brass sample (sample A) after 307 min of creep

showing a selected slab. The temperature increases from the top to the bottom.

IV. Results 53

Applying Eq. 2.1 to two slabs one located at a reference temperature (usually the

smallest temperature in the analyzed region) and the second at temperature T, the ratio of the

corresponding strain-rates can be written as follows:

0T

4.1) ⎟⎟⎠

⎞⎜⎜⎝

⎛⋅⋅

Δ⋅=

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛⋅⎟⎟

⎠

⎞⎜⎜⎝

⎛⋅

TTRTQ

TGTG

tTtT

tTtT

nn

0a

0

0

0 )()(

),(),(

),(),(ln

σσ

εε&

&,

where . The local strains (and strain-rates) are obtained from tomographic

reconstructions performed at the reference time and current time t. Assuming that the

volume of the creeping material is constant the local strain in the slab can be related to the

local change in sample’s cross section as:

0TTT −=Δ

0t

4.2) ⎟⎟⎠

⎞⎜⎜⎝

⎛=

),(),(

ln),( 0

tTAtTA

tTε ,

while the local strain-rate in the time interval 12 ttt −=Δ (and associated to time ) is given

by:

2t

4.3) ⎟⎟⎠

⎞⎜⎜⎝

⎛Δ

=⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟⎟

⎠

⎞⎜⎜⎝

⎛Δ

=Δ−

=),(),(

ln1),(),(

ln),(),(

ln1),(),(),(

2

1

1

0

2

0122 tTA

tTAttTA

tTAtTAtTA

tttTtT

tTεε

ε& ,

where is the average area of the undamaged material (without voids) in the slab at

temperature T and time t. It is important to note that the evaluation of the strain-rate at time

does not require the measurement of the initial state (at time ) but only the state at time .

However, if the local creep curves are of interest then the reference state is also required (this

will be shown later).

),( tTA

2t

1t0t

Figure 4.2 exemplifies the position change of three slabs during creep

and the methodology used for strain evaluation.

Considering a sample with constant cross-section in the reference state it becomes

possible within certain conditions to evaluate without knowing the stress exponent n. The

method applied to a constant cross sectional area will be exemplified for the evaluation of

in brass (sample A) in section

aQ

aQ

4.1.3. Usually in practice not only volumes with constant cross-

IV. Results 54

section are reconstructed, but also regions with varying section (see Figure 4.2) where the

stress is slightly changing. The stress ratio in Eq. 4.1 can be expanded in a series of the strain

difference between slabs at temperatures T and from which keeping only the first terms we

get:

0T

4.4) )),(),((1n

≅),(),(

),(),(

00

0 tTtTntTAtTA

tTtT

n

εεσσ

−+⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡ ,

This equation suggests that the stress ratio approaches 1 if the strain difference between

slabs is small. This condition is usually fulfilled in case of small temperature gradients, when

the ratio of the shear moduli approaches unity, too. Considering the stress exponent n to be

known, Eq. 4.1 suggests that can be determined even in such cases, however, only with

the restriction that the cross-sectional area of the slices does not vary strongly within short

time intervals and the assumption of a uniaxial stress state is approximately fulfilled. In case

of the present experiments the stress triaxiality as determined from FE calculations in the

investigated region with varying cross section was below 0.38.

aQ

Figure 4.2. Examples of slab displacement along the tensile axis and the methodology of

strain and strain-rate calculation for the slab. N represents the number of slices in the slab.

IV. Results 55

4.1.1 Temperature distribution calibration

The temperature distribution in the specimen was determined by finite element (FE)

calculations taking into account the accurate shape of the sample and the temperatures

measured by the thermocouples. The calculations were performed for thermal steady-state

using the following material constants for brass (sample A): heat conductivity λ = 115

W/m K, specific heat c = 380 J/kg K and mass density ρ = 8530 kg/m3. For steel (sample B

and C) the material constants used were: heat conductivity λ = 24.2 W/m K, specific heat c =

460 J/kg K and mass density ρ = 7650 kg/m3. The FE results were confirmed by temperature

measurments performed by thermocouples welded at different hights for all samples. Figure

4.3 shows the temperature distribution in samples A (brass), B (steel), and C (steel) as a

function of the distance from the center of the specimens. There is a non-linear temperature

variation in regions with varying cross section, while in the middle of the samples, where the

cross-section is practically constant the temperature changes linearly with distance. Based on

temperature profiles obtained from FE simulations a certain average temperature and a

corresponding tolerance can be now ascribed to each slab. It will be shown later that the

temperature change of one slab during the short strain interval considered for the evaluation

of is small and the error in temperature is mainly determined by its variation over the

slab’s height.

aQ

IV. Results 56

Figure 4.3. Temperature profiles (a) and contour plots (b) along the stress axis as obtained

from FE simulations for samples A (brass), B (steel) and C (steel). Lines and number in (a)

indicate the linear temperature gradient at the center of the samples.

4.1.2 Image correlation for evaluation of tomographic data

Considering the tomographic reconstruction similar to data delivered by a 3D

extensometer it becomes possible in case of in situ tests to trace the movement of each

material slice or volume. The cross-correlation method explained in section 3.4.3 was used for

that purpose. Once the displacement of a slice was found, the corresponding local true strain

could be calculated as shown in Eq. 4.2 and depicted in Figure 4.2. For higher accuracy an

average strain over 10 slices making up a slab was calculated. Selecting a slab as structural

IV. Results 57

unit for evaluations was imposed also by the nearly constant temperature condition required

in the slab in the time interval selected for evaluation. Usually the difference in temperature

between the lower and upper slice in the slab is larger than the average temperature change of

the slab over the selected time interval. Figure 4.4 shows for example the displacement of

three selected slices (with regard to their original position) during the creep test. The material

slices were selected at the top, in the middle and at the bottom of the volume measured and

correlated for the whole creep test (steel, sample C). Since the bottom of the sample was kept

fixed all material slices move upwards. A slice at a higher position (lower temperature)

experiences therefore a larger displacement since the displacements of all slices situated

bellow are added together. Since the cross section of the material slightly varies along the

vertical axis ( ) the displacements of the slices are different and characteristic

for the local inhomogeneous deformation. For example, in the interval from 234 min to 300

min the slice 70 (

19.1/ minmax =dd

Figure 4.4) experience a temperature change of ~0.68 K which is small

compared to the total temperature difference in the analyzed volume of about 50 K.

Figure 4.4. Displacement of three slices along the loading direction as obtained from cross-

correlation of tomographic slices. Sample C (steel).

IV. Results 58

4.1.3 Method for aQ evaluation on samples with constant cross section

The structure of the sample A (brass) before fracture is shown in Figure 4.5a), where the

gradual increase of damage with increasing temperature is evident. Most quantitative

evaluations were performed on the central part of the sample, which fulfilled the constant area

criterion required by the constancy of the temperature gradient and stress. The selected ROI

indicated in dark grey (Figure 4.5a) was a cylindrical volume of 480 µm (300 voxels) in

height and of almost constant cross-sectional area (relative deviation less than 0.5%). Figure

4.5b) shows the spatial distribution and evolution of cavities within a subvolume of the ROI

after creep times of 52 and 110 min. Compared to samples tested in the laboratory, the

diameter of the in situ investigated specimen was small (~1 mm); however, the resolution of

the tomographic reconstructions (~0.002 mm) yields reliable local geometric information for

calculation of local strains [154] and consequently of local creep curves for different

subregions of the sample.

Figure 4.5. (a) Tomographic reconstruction of the sample A (brass) after 440 min creep. The

darker region indicates the selected ROI. (b) Spatial distribution and evolution of pores within

a subvolume of the ROI after creep times of 52 and 110 min.

IV. Results 59

4.1.3.1 In situ creep curve of brass (samples A)

The overall creep curves for sample A (brass, see Table 3.2) is plotted in Figure 4.6a),

together with the local creep curves of the material slabs. The black solid line is the curve

calculated from the total displacement of the sample (during the in situ measurement)

considering an effective length of 1.8 mm as defined in section 5.1.1. Local creep curves for

nine slabs (one slab was obtained as an average over 10 slices) positioned 96 µm from each

other along the tensile axis are shown in Figure 4.6a). The curves for the slabs begin at

52 min, when the first deformed state was reconstructed, and show a quite different behavior

as a function of time. The curve associated with the slab at the lowest temperature of 611 K

shows a nearly constant slope, characteristic of steady-state creep, while the slab with the

highest temperature of 641 K experiences the transition from steady-state creep to accelerated

creep; their average creep-rates at 52 min differ within a factor of 2, they are ~1.5·10-6 and

~3.0·10-6 s-1, respectively. It is important to mention that the local steady-state creep-rates in

the ROI of the pre-deformed sample are about one order of magnitude higher than the steady-

state creep-rate of large samples without pre-deformation and tested under similar conditions

in the laboratory [58].

Figure 4.6b) shows the global strain-rates as a function of time for sample A, with the

typical behavior of a dual phase α/β brass. After the “pseudo-minimum” Iε& the strain-rate

oscillates around a more or less constant value (as obtained by averaging the data in the

corresponding interval and shown in red) until a time of about 200 min, followed by tertiary

creep. Due to oscillations it is difficult to see that creep of the measured tomographic sample

has reached the steady state or not. However, from the point of view of the results presented

here, which correspond for the selected slabs this has no significance. The creep curves of the

slabs restricted to the ROI (Figure 4.6a) clearly show (lower curves, maximum T=628K) that

the local strain is linear with time.

Similar creep curves for a brass alloy were shown by Willis and Jones [141]. They

showed creep curves, which were generally tertiary-stage dominated, and are very similar to

the creep curve presented here.

IV. Results 60

Figure 4.6. (a) Local creep curves of slabs at different temperatures compared to the creep

curve of the tomographic sample A recorded during in situ measurement. (b) Variation of

strain-rate with time during creep of Cu-40Zn-2Pb at 25 MPa and 673 K at the bottom of the

sample.

4.1.3.2 aQ values according to literature and tomographic method

According to the data in Figure 4.6, steady-state creep prevails in the slabs of the ROI in

the time interval 52–137 min. In the particular case in which a volume with constant cross

section is analyzed the ratio of stresses and shear moduli in Eq. 4.1 approaches unity if the

temperature gradient between slabs is small compared to the applied temperature. For these

IV. Results 61

cases, the logarithm of the normalized local strain-rate referring to the lowest temperature, 0T ,

becomes a linear function of 0TTT −=Δ . Eq. 4.1 can be rewritten as:

TRTQ

TT a Δ≈ 2

00 )()(ln

εε&

& 4.5)

or the states investigated in sample A the stress ratio (given by Eq. 4.4 has only a

secon

F

d-order contribution and can be considered equal to 1. Figure 4.7 shows the normalized

strain-rates as a function of TΔ . Both data sets for time intervals 52–110 min (at 52 min in

Figure 4.7) and 110–137 m (at 110 min in in Figure 4.7) fit Eq. 4.5 with regression

coefficients of 0.89 and 0.93, respectively. The normalized strain-rates at 52 and 110 min

yield apparent activation energies for steady-state creep of 101 ± 12 and 117 ± 12 kJ/mol,

respectively, which can be considered equal within the limits of their standard deviations

considered as errors. Their average value of 109 kJ/mol differs by about 20% from the value

of 133 ± 9 kJ/mol obtained from laboratory tests on large samples with somewhat different

chemical composition (CuZn36Pb2.5) [58].

igure 4.7. Logarithm of average strain-rate ratios at temperatures T and T0 = 611 K at two F

creep times (52 and 110 min) as a function of temperature difference TΔ (the error of the

strain-rate ratios was estimated from the standard deviation of the area ch ge between the 10 an

cross-correlated slices making up the slabs; their relative error is about 15–20%).

IV. Results 62

4.1.3.3 Damage development in brass (samples A)

The temperature dependence of damage can simply be characterized by calculating the

area fraction of voids on slices perpendicular to the temperature gradient. The results

show

AA

n in Figure 4.8 indicate an almost constant AA in the initial stage (and also at 52 min),

which is followed by a progressive increase as a function of temperature and time, A

be

A

becoming as large as 7% before final rupture. According to Figure 4.8b (corresponding to the

ROI) the average AA

ch

stant along the sample length with an average area fraction of 0.019%. There exist,

howe

– disregarding its fluctuations – increases with temperature. It should

noted that in the present experiment voids are observed only after nucleation and growth,

once they have rea ed the resolution limit. Thus there are two possible contributions to the

observed inhomogeneous void-area increase: (i) voids at higher temperatures grow at higher

rates and (ii) in material regions with higher temperatures, voids have nucleated at higher

rates.

Despite of the resolution limit mentioned above the damage in the initial state seems to

be con

ver, regions in which damage is more localized, Figure 4.8b. After 52 min creep, the

average area fraction is 0.024% and it shows a faster development towards higher

temperatures. It should be also noticed that the initial distribution of the damage plays an

important role in the final fracture [155], since practically all the regions with a higher area

fraction of cavities developed faster than the less cavitated ones at all temperatures. It is

observed that even small variation in AA in the initial state can lead to large differences in

damage accumulation at the end of the creep lifetime. Even in the case when a temperature

gradient is applied, the influence of the itial damage on its further development is observed

over the whole temperature range. In the case of brass the regions where the A

in

A has

developed more rapidly can be already recognized after 137 min of creep.

IV. Results 63

Figure 4.8. (a) Variation of the area fraction of voids in slices perpendicular to the

temperature gradient as a function of temperature at various creep times (b) Enlarged view of

the marked region in (a).

AA

IV. Results 64

4.1.4 Method for aQ evaluation on samples with varying cross section

The structure of sample B (steel, K985 MPa,180 b == Tσ ) and C (steel,

K1013 MPa,210 b == Tσ ) after 951 and 194 min of creep is shown in Figure 4.9a) and b),

respectively. The correlated volume at these two states (region of interest -ROI-) is depicted

by the dark grey region in the middle of the volumes. The quantitative evaluations were

performed on the ROI. It can be observed in Figure 4.9a) and b) that the cross section slightly

varies from top (lower temperature) to bottom (higher temperature) for both samples,

however, for sample B the variation is less significant. The selected ROI for samples B and C

has a height of 699 µm (437 voxels) and 672 µm (421 voxels), respectively.

Figure 4.9. Tomographic reconstruction of the sample B (steel, K985 MPa,180 b == Tσ )

after 951 min creep (a) and of the sample C (steel, K1013b MPa,210 == Tσ ) after 194 min

creep (b). The darker regions indicate the selected and cross correlated regions of interest.

4.1.4.1 aQ values according to classical and tomographic methods

The creep behavior of the AISI 440B steel was first investigated by the conventional

method on large specimens. The stress exponent was evaluated from the slope of n )ln( minε&

vs. )ln(σ obtained from tests at constant temperature 873 K (see Figure 4.10a). Additionally,

IV. Results 65

the activation energy corresponding to the minimum creep-rate was obtained from tests

performed at different temperatures for two different normalized stress values of 1.4·10-3 and

3.1·10-3 (σ of about 77 MPa and 180 MPa, respectively). The corresponding linear fits

performed on the plot of )ln( minε&⋅− R

aQ

as function of yielded the values of

kJ/mol and 389±17 kJ/mol for the small and large normalized stress,

respectively. This means that does not depend on stress in the studied interval, which is in

good agreement with literature data for stainless steel [

T/1

15±388a =Q

45].

Figure 4.10. Evaluation of the stress exponent (a) and activation energy of steady-state

creep (b) from laboratory tests according to the conventional method.

n

IV. Results 66

4.1.4.2 In situ creep curves of steel (samples B and C)

The overall creep curves for samples B (steel, K985 MPa,180 b == Tσ ) and C (steel,

K1013 MPa,210 b == Tσ ) are plotted in Figure 4.11a) and c), respectively, together with

the local creep curves of the material slabs. The thick solid lines are the curves calculated

from the total displacement of the samples (during the in situ measurement) considering an

effective length of 1.8 mm as defined in section 5.1.1. The local curves (gray lines with

symbols) were obtained from tomographic evaluations by averaging the local strain over the

slices making up the slab. The reference state for calculation of the local creep curves was

considered at time 852 min and 156 min for sample B and C, respectively. These points on

the creep curve coincided with the beginning of the secondary state. For easier comparison

the local creep curves where shifted vertically to bring them together with the overall curves.

The insert in Figure 4.11a) serves this purpose, too. 44 and 42 slabs (each slab obtained as an

average of 10 slices) were followed from the beginning of the secondary state until the

beginning of the tertiary state for sample B and C, respectively. For a better representation,

only local creep curves for fifteen and sixteen slabs separated 48 µm from each other are

depicted in Figure 4.11a) and c) for sample B and C, respectively. For sample B, the average

creep-rates of the local creep curves differ by a factor of about 8 varying between

and for the coldest slab and hottest slab, respectively. The global

strain-rate was determined by hotter regions not captured by tomography. For

sample C, the average creep-rate for the local creep curves differ within a factor of about 11

and they lie between and for the coldest and hottest slabs,

respectively. The global strain-rate was .

-18 s100.8~ −⋅ 105.6~ ⋅

6 s102.1~ −⋅

.1~

-17 s−

-1

101⋅ -16 s− -15 s−

-1

102.1~ ⋅

6 s103. −⋅5~

The different creep behavior of the slabs is primarily due to the temperature gradient

leading to inhomogeneous strain distribution in the sample, i.e. the local creep behavior of

material slabs will differ from the overall behavior of the sample. Figure 4.11b) and d) show

the global strain-rates as a function of time for samples B and C, respectively. Sample B

experiences a long primary creep regime in which the strain-rate decreases continuously

reaching a steady-state after about 15 hours. For sample C, deformed at higher stress and

higher temperatures, a short transient regime with increasing and decreasing strain-rate is

observed between 60 and 120 min of the creep test. Such transient was also observed during

creep of a small brass sample [156]. All our evaluations were focused on the secondary

IV. Results 67

regime, when a nearly steady-state in the whole sample and in local material slabs was

reached.

It continues in the next page

IV. Results 68

Figure 4.11. Overall creep curves for samples B and C (thick solid lines) together with creep

curves of local material slabs (gray lines with symbols) obtained from tomographic

evaluations (a) and (c). Overall creep-rate vs. time for sample B (b) and C (d).

IV. Results 69

For the analysis of tomographic data it is preferred to plot the left hand side of Eq. 4.1

against )( 0 TTRT ⋅⋅Δ . When TTT ≅<<Δ 0 then the left hand side of Eq. 4.1 becomes a

linear function of TΔ and the fitted line should intersect the origin, too. Figure 4.12a) and b)

show the variation with temperature of the parameter-ratios entering Eq. 4.1 namely, that of

the strain-rates, stresses and shear moduli. It is evident that the main contribution to comes

from the ratio of the strain-rates and the other two should be considered as correction factors

with decreasing importance. Neglecting for example the shear modulus correction, has an

effect of less than 5% on . The slopes of the fitted lines for sample B and C (

aQ

aQ Figure 4.12a

and b) fit Eq. 4.1 with regression coefficients of 0.988 and 0.995, respectively, yielding

apparent activation energies equal to 377 ± 30 kJ/mol and 402 ± 12 kJ/mol for samples B and

C, respectively. These values are equal within experimental error with each other and with the

result obtained according to the conventional method. It should be noted that the error of

can be further reduced by increasing the resolution of the tomographic reconstruction. For

high energy beams, however, detectors with better efficiency are needed to pass the actual

resolution of about 2 µm.

aQ

IV. Results 70

Figure 4.12. Linear regressions performed on tomographic data to obtain ; (a) sample B

and (b) sample C. The diagrams also show the contributions of different ratios entering

Eq.

aQ

4.1. The sum of the three functions gives the left hand side of Eq. 4.1.

4.2 Microtomographic investigation of damage in E911 steel after long term creep

This part of the thesis is focused on the microtomographic evaluation of damage

distribution along the notch radius as well as cavities shape. The 3D shape, orientation and

size distribution will be described. Results from conventional metallography and tomography

IV. Results 71

will be compared. Also, a comparison of local damage distribution with the distribution of

selected stress parameters as obtained from FE simulation will be performed.

4.2.1 Conventional damage evaluation by OM

The cylinder extracted from the notch region of the hollow cylinder sample (E911 steel)

shown in Figure 3.4 was investigated by conventional optical microscopy (OM). Cavities

were analyzed on 2D sections perpendicular to the notch radius at three different distances

from the inner surface. Figure 4.13 shows examples of findings by optical microscopy. At the

magnification of the pictures, the visible cavities show an irregular shape, typical for large

cavities. From the 2D sections, however, it cannot be stated if the irregular cavity shape is due

to coalescence or to the growing mechanism. Also the coalescence of cavities of different

sizes could be observed (middle surface).

Figure 4.13. Micrographs of cavities at the different cross sections of the specimen (polished

surface).

IV. Results 72

The microstructure of the E911 steel is shown in Figure 4.14. It consists of tempered

martensite, the primary austenite grains are still visible. The cavities however are linked with

martensite laths. It is obvious that the shape of the cavities is influenced by the random

orientation of the single martensite laths in their direct vicinity.

Figure 4.14. Optical image of the E911 steel after 26,000 h of creep (uniaxial load of 8543 N

and internal pressure of 17.5 MPa) showing a martensitic microstructure. The voids are

mainly aligned with the martensite laths.

The evaluation of cavity density, area fraction and size was done by means of Digital

Image Processing (DIP). The results are given in Table 4.1.

Distance from

inner surface

Size (μm) Number

(1/mm2)

Area fraction

(%) Minimum Maximum Equivalent

diameter

400 μm 0.30 6.65 1.48 861 0.22

1000 μm 0.27 8.52 1.50 1848 0.53

1600 μm 0.21 8.36 1.77 2101 0.73

Table 4.1. Evaluation of cavity size, equivalent diameter, cavity density and area fraction at

three different distances from the inner surface.

IV. Results 73

4.2.2 Tomographic evaluation of damage

The microtomographic evaluation was focused on damage distribution along the notch

radius as well as on cavity shape. In the investigated state the voids appear to be

homogenously distributed on slices perpendicular to the notch radius (Figure 3.6), but their

number and average size varies along the radius (Z direction). Figure 4.15 shows qualitatively

the increase of pore density from the inner side towards the outer surface of the notch.

Figure 4.15. Tomographic reconstruction of the E911 steel specimen after 26,000 h of creep.

The section reveals inhomogeneous distribution of cavities along the notch radius (Z

direction).

Figure 4.16 shows two sub-volumes containing some local accumulation of voids of

various sizes probably situated at grain boundaries, information impossible to obtain from

tomography. They are located in the highly damaged region of the sample, where also the

largest pores were found.

IV. Results 74

Figure 4.16. Two sub-volumes taken from the region of the sample with maximum damage

(at 1600 µm from the inner surface of the notch) showing a region with high void density.

To compare the tomographic and metallographic data the cavity density on 2D slices

was first evaluated. Figure 4.17a) shows the variation of this density along the notch radius.

Each point corresponds to one slice with thickness equal to the voxel size of the

reconstruction (0.33 µm). An abrupt increase1 in cavity density is observed at about 200 µm

from the inner surface of the notch (where the pressure was applied). The cavity density

increases monotonically and reaches a maximum at about 1600 µm after which a slight

decrease towards the outer notch surface is observed. The maximum density is about

1700 pores/mm². The area fraction of voids (area of pores/total area investigated) shows

similar distribution as the cavity density and has its maximum value of 1.5% also around

1600 µm (Figure 4.17b). A fair agreement between tomographic and metallographic results

was obtained, the metallographic pore densities being somewhat larger.

1 An influence of the pressurizing medium (air) could be assumed: at the inner surface higher

contents of C and N have been found.

IV. Results 75

Figure 4.17. Cavity density (a) and area fraction (b) of cavities along the notch radius. The

results shown in three different gray levels were obtained from three tomographic

reconstructions coupled into one large volume. The circles show the cavity density obtained

from metallography.

The 3D tomographic reconstructions allowed the analysis of cavity shape, their

orientation and spatial distribution. Cavity shape was characterized by a complexity factor

( ) related to the exclusion volumes resulting from the intersection of the real cavity with

its equivalent ellipsoid as defined in section

CF

3.4.2. The varies between zero and two and

offers a good possibility to distinguish between regular cavities and those affected by

coalescence.

CF

The distribution of the complexity factor (CF ) and elongation ( e ) as a function of

volume is presented in Figure 4.18. Both the CF and e decrease at small cavity volumes

IV. Results 76

reaching a minimum value, followed by a further increase. This effect is related to the cavity

size and the error introduced when the equivalent ellipsoid is fitted to the cavity. This error

decreases with the increase of cavity size. However, as it is improbable that cavities with a

small size are affected by coalescence, these cavities will be considered as non-coalesced ones

for the calculation of the cavity size distribution function. For the calculation of cavity

orientation, a minimum cavity of 125 voxel and a minimum elongation of 1.4 will be

considered. With further increase of cavity size (from 125 to about 3000 voxel) both CF and

increase. For cavity sizes larger than 3000 voxel a large scatter is observed for both

parameters (CF and ) which is due to the smaller number of cavities sizes in that range.

e

e

Figure 4.18. Complexity factor CF , and elongation , as a function of the cavity volume.

The bin size for the volume was 20 voxels.

e

Based on a visual analysis and on the distribution functions of and , cavities with

have been considered regular and those with as complex.

CF

CF

e

5

5.0≤CF 5.0>CF Figure 4.19a) to

d) show selected cavities and their equivalent ellipsoids. Figure 4.19a) shows a very large

pore of 13,679 voxels which has experienced coalescence with several smaller pores located

along a given direction. Figure 4.19b) illustrates a case when the , however, it is

visible that the cavity is affected by coalescence indicating that the CF criteria alone may fail

in cases of coalesced cavities with broad interconnecting bridges (indicating that the

coalescence process is not recent).

.0≤

Figure 4.19c) shows the case of a recent coalescence where

IV. Results 77

the interconnecting bridges are narrow, hence the CF is larger than 0.5. Figure 4.19d)

represents the most typical case, a pore with a 5.0≤CF and 9.1≤e .

Figure 4.19. a) to c) Cavities with volume larger than 3000 voxels and their equivalent

ellipsoids. Volume, complexity factor (CF ) and elongation ( e ) are indicated in the picture. d)

Spheriod cavity shape typical for non-coalesced cavities.

Since coalescence generally leads to long cavities a new criteria based on the aspect

ratio of the equivalent ellipsoid was introduced. From the definition of elongation of the

cavity presented in section 3.4.2 the value of 9.1max =e was selected on a visual basis to

discriminate between coalesced and non coalesced cavities. Finally, a population of about

69,100 regular cavities could be separated, which fulfilled the condition to be affected only by

nucleation and growth. The histograms of voids major axis and equivalent radius are

shown in

1a ea

Figure 4.20 together with a fitted distribution function proposed by Riedel [13]

(presented in section 2.5.4).

IV. Results 78

The parameters βα ,,, 21 AA and γ of the size distribution function (Eq. 2.19) can be

related to general power-law functions describing non-stationary laws for the growth rate of

cavity radius a& nd the nucleation rate *J as expressed by Eq. a

o

2.17 and 2.18 [13]. According

to Figure 4.20 the function defined by Eq. 2.19 describes well the distribution of cavity sizes,

however, it approximates better the distribution of the equivalent radius ea than that f the

major axis 1a (only f results for ea will be discussed in the following). Since only one

deformation state was investigated the time t in Eq.

it

2.19 was considered constant, which

permitted the determination of exponents α and β characterizing cavity growth. The value

of 05.95.1= 0±β is close to 2 as predicted by the constrained diffusion mechanism [12,13].

The fitted exponent 200)( 1/() −+ ≅αγα is a very large number, which in case of realistic α

and γ of the order of unity, suggests 1≅α .

Figure 4.20. Probability density functions of cavity’s equivalent radius ea and major

ellipsoid axis 1a . The continuous lines represent the fit of Eq. 2.19. Only c with

5. and 9.1≤e were considered (cavities before coalescenc

avities

e happened). 0≤CF

The orientation of cavities with 5.0≤CF has been characterized based on the direction

of the eigenvector pointing along the major axis of the equivalent ellipsoid. For this purpose

the usual spherical coordinates, the polar angle θ and the azimuthal angle ϕ were used.

Figure 4.21 shows a system of curvilinear coordinates that are natural for describing positions

IV. Results 79

on a sphere. θ was defined with respect to the Z axis of a Cartesian system pointing along

the radius of the notch. The polar angle θ varies between πθ ≤≤0 . The azimuthal angle ϕ

varies between πϕ ≤≤0 in the xy-plane from the x-axis.

Figure 4.21. Definition of spherical coordinates used for describing the spatial orientation of

creep cavities.

The azimuthal angle ϕ , did not show any preferred orientation as a function of the

distance from the inner surface ( Z ) or cavity volume. Selecting a bin size of °=Δ 5ϕ the

corresponding number histogram for ϕ showed a nearly constant behavior. This means that

the voids are randomly oriented on planes lying perpendicular to the notch radius. These

planes (X-Y) contain, however, the direction of the applied axial load. On the other hand the

distribution of the polar angle θ , shows a peak at 90° with respect to the Z axis (Figure

4.22). This maximum is characteristic for almost all θ distributions calculated for

subvolumes situated at various distances from the inner surface. The maximum of the

distributions becomes sharper with increasing Z .

IV. Results 80

Figure 4.22. Distribution of the polar angle θ at different Z locations along the notch radius.

Z bin size = 33 µm (100 slices), not all the curves are plotted. θ bin size = 5°. Selection

cavity condition: 5.0,9.11 vox,125umecavity vol min4. ≤≤≤≥ e CF .

V. Discussion 81

CHAPTER V

DISCUSSION

5.1 Evaluation of apparent activation energy of creep, aQ

5.1.1 Comparison of strain distribution in the real sample and FE models

The overall strain is a macroscopic quantity characteristic for the whole sample. Due to

the restricted field of view of the CCD camera, tomography gives information only from a

portion of the gauge length and the average “tomographic” strain might not coincide with the

strain calculated from the measured displacement of the sample’s top (Figure 4.11a, c and

Figure 4.6a). In order to check if the local strain distribution delivered by tomography

matches the strain distribution in homogeneous material, FE simulations have been

performed. The comparison is shown in Figure 5.1a) and b) for the sample B and C,

respectively. Similar strain profiles were obtained for sample A with the maximum of the

strain distribution function closer to the center of the sample due to the smaller temperature

gradient applied. Therefore, the discussion will be focussed on sample B and C. The

conclusions are, however, general and applicable also to sample A. The contour plots and the

above located diagrams in Figure 5.1 clearly show that the maximum strain regions are shifted

towards higher temperatures (negative position values in Figure 5.1a) and b) which were not

captured by tomography. The regions evaluated are indicated by vertical lines and are

confined to the interval [-0.67, -0.05] mm for sample B and [-0.90, -0.41] mm for sample C.

The inserts in the upper right corner of the figures show the strain distribution obtained from

tomography at different creep times. The continuous black line with full squares depicts the

distribution at the end of steady-state. The behavior of material slabs at different positions

with respect to the maximum of the strain distribution is reflected in the local creep curves

shown in Figure 4.11a) and c). Hence a slab situated closer to the maximum of the strain

distribution has not only a larger strain, but also elongates at a larger strain-rate. In FE

simulations a Norton law describing steady-state creep with appropriate constants for the

investigated steel was used and deformation was simulated until the elongation of the sample

reached the value describing the end of secondary stage. The strain distribution obtained from

FE simulations facilitates a better understanding and design of the in situ experiment. It

becomes evident that the field of view of the CCD should be positioned in the sample region

V. Discussion 82

with the highest strain gradient, where the ratio of strain-rates )),(/),(( 0 tTtT εε && is the largest

and less affected by experimental errors.

The strain distribution obtained from FE calculations allows introducing an “effective

length”, a quantity that facilitates the calculation of the average strain and the overall creep

curve. In case of tests performed at constant temperature the strain distribution along the

sample axis can be well described by a Gaussian function. For asymmetric heating the

distribution becomes asymmetric, which suggests selecting a criterion based on the relative

amount of strain present in a given volume around the center of the strain profile. The average

strain in the sample of length is obtained as follows: L

5.1) L

Udzz

Lexp)(1

== ∫εε . (14)

where the integration is performed over L and is the measured displacement. Since the

strain is localized in a small region, using L as the effective length is not adequate. Therefore

we define an effective length equal with the interval around the center of the strain

profile, which contributes by 95% to the total elongation of the sample. The was

evaluated from strain profiles obtained from FE simulations and its value is about 1.8 mm,

almost twice the gauge length of the sample with constant cross section. The average strain

(and strain-rate) shown in

expU

effL

effL

Figure 4.11 (thick solid lines) were calculated according to Eq. 5.1

considering . It should be noted, however, that the value of is not affected by

the selected , which affects only the overall curves.

mm8.1=L

effL

aQ

V. Discussion 83

Figure 5.1. Strain distribution (obtained from FE simulations) along the load axis of the specimens characterizing the end of steady-state (a) sample

B and (b) sample C. The upper part in each figure shows the asymmetric strain distribution. The volume measured by tomography is marked by

vertical lines. The inserts in the upper right corners show the strain distribution in the sample at stages where the tomographic reconstructions were

performed.

VI. Discussion 84

5.1.2 Error estimation

For every new evaluation method it is important to characterize the sources of error

influencing the final results. The resolution of the tomographic reconstruction is of about

2 µm, which in case of a sample with diameter equal to 1 mm, gives a relative error of the

local strain and strain-rate below 1% (the uncertainty in determining the diameter is

considered equal with the tomographic resolution). This is much smaller than the standard

deviation of the average strain (and strain-rate) of one slab, of about 12%, determined by the

different creep behavior of the slices making up the slab. The stress ratio in Eq. 4.1 reduces

also to the ratio of two areas and consequently it has the same relative error.

Another source of uncertainty is related to the local temperature calibration obtained

from FE simulations. Selecting a slab (instead of a slice) as structural unit the error of the

temperature associated to the slab depends on the location of the slab (through the local

temperature gradient and local displacement) and the number of slices included. In case of

samples B and C the slabs were considered to be made up of 10 slices, which gave maximum

temperature errors of ±0.8 K and ±0.9 K, respectively. The temperature error due to sample

elongation during steady-state creep was estimated based on the experimentally detected

displacement for the uppermost slab in the reconstruction and the temperature gradient in the

specimen at the beginning of steady-state. The total analyzed creep strain in sample B was

2.3% ( =310 min) and 5.5% ( = 174 min) in sample C. These led to a temperature change

of the uppermost slab of about 0.3 K and 1.5 K for specimens B and C, respectively. Since the

slabs are moving upwards the temperature error becomes asymmetric. The total temperature

errors are equal to +0.8 K and -1.1 K for sample B and +0.9 K and -2.4 K for sample C. These

errors correspond to the slab with the highest location and compared to the total temperature

difference in the analyzed volume of about 50 K are small. The acceptable accuracy of the

local temperature and strain-rate ratios, as well as the large number of slabs analyzed, makes

an accurate evaluation of the activation energy possible. Both values of obtained from in

situ tests as well as the values obtained from conventional laboratory tests are equal within an

error of 5%.

tΔ tΔ

aQ

VI. Discussion 85

5.2 Damage investigation in E911 steel

Damage dependence on different stress parameters under multiaxial creep was usually

studied on notched specimens [73] and the stress parameters (maximum principal stress,

equivalent von Mises stress) used for interpretation of damage were characteristic for the

entire sample. Nix et al. [135] have also considered the principal facet stress prevailing at the

center of the notch, their argument being that the stress state around the notch center remains

constant in a reasonably large volume. As shown in Figure 4.17 microtomography is able to

characterize damage at the mircometer scale giving not only an average value, but also the

distribution of it along the notch radius. Since the stress state varies along the radius the

availability of local damage distribution offers a more accurate possibility to check the

influence of different stress parameters on creep damage. The local stress and stress state were

obtained from finite element (FE) simulations considering the viscoplastic Norton-law to

describe creep of the notched hollow cylinder. However, considering only steady-state creep

means that strain hardening effects are neglected. The experimentally determined parameters

used in the creep law were the stress exponent n = 7 and the apparent activation energy

. Elastic properties of the E911 steel were taken into account by considering a

Young’s modulus of 177 GPa and a Poisson’s ratio of 0.31. A constant load of 8543 N and a

pressure of 17.5 MPa were applied at one extreme of the sample (along the Y axis) and at the

inner surface of the hollow cylinder, respectively. After loading four stress parameters were

calculated at each integration point: the maximum principal stress (MPS), the equivalent von

Mises stress (VMS), the principal facet stress (PFS) and the stress triaxiality. The distribution

of the MPS, VMS and PFS in the notch after steady-state creep deformation of 0.30%

corresponding to a creep time of 20,000 h are shown in

kJ/mol458a =Q

Figure 5.2 (0.32% in 6,000 h

correspond to primary creep and were not considered for the calculation of the stress

parameters). Although the distributions of the MPS and VMS are different, their values are

nearly of the same order of magnitude, all increasing towards the outer surface of the notch.

The PFS, however, has an average value higher than the VMS and MPS by a factor of about

1.9 and it doesn’t show a monotonic increase towards the outer surface of the sample.

VI. Discussion 86

Figure 5.2. Stress distribution inside the notch after 20,000 h of creep ( %30.0=ε ). The

sample region measured by tomography is marked by the dotted line.

According to the FE results neglecting transient creep should have only a minor effect

of the final stress distribution in the notch. The contour-plots of the stress parameters evolve

differently during the first 8,000h. For example, the position of the kink of the MPS and PFS

(see Figure 5.4b and c) varies during this time from Z = 1900 µm until 1350 µm, as shown in

Figure 5.3. However, after a creep time of 8,000 h the contour-plots of the stress parameters

evolve in a nearly self-similar manner, i.e. the highest and lowest values remain practically

unchanged and in the case of MPS and PFS the location of kink position remains between

Z = 1350 µm and 1250 µm for over 12,000 h.

VI. Discussion 87

Figure 5.3. Variation with time of kink location of the MPS and PFS parameters (Figure 5.4b

and c).

The distributions of the stress parameters along the notch radius after 20,000 h of creep

are plotted in Figure 5.4a) to d) and compared with the cavity density distribution. The VMS

and MPS (Figure 5.4a and b, respectively) have similar values, however, their distributions

are quite different. The VMS increases continuously along the radius, while the MPS shows a

kink at Z ≈ 1255 µm. The PFS shows also a kink at the same position which is, however, the

minimum of the distribution.

To obtain a more accurate damage parameter the volumetric cavity density

corresponding to local volumes made up of 30 slices was evaluated. The result shown in

Figure 5.4a) to d) is, however, similar to the distribution obtained from 2D sections. None of

the parameters selected is 100% similar to the cavity density, which can have two reasons: a)

Due to its limited resolution, of about 0.5 μm, microtomography does not detect all cavities.

This hypothesis can, however, be dropped since the modus of cavities size distribution was

captured by the technique (Figure 4.20), suggesting that the proportion of non-detected

cavities (assuming unimodal distribution) is small, which will not change significantly the

found distribution. b) The stress parameters selected does not influence equally the

mechanisms of nucleation and growth, which both contribute to the observed distribution.

An eventual correlation between stress parameters and creep damage can be obtained

based on the comparison between the corresponding distributions (Figure 5.4a to d). The

cross-correlation coefficients of the selected stress parameters and the volumetric cavity

density are given in Table 5.1.

VI. Discussion 88

Cross-correlation Coefficient

Stress triaxiality

factor, ft

Maximum Principal

Stress, MPS

Von Mises

Stress, VMS

Principal Facet

Stress, PFS

Cavity density

function 0.978 0.917 0.872 -0.594

Table 5.1. Cross-correlation coefficients obtained from comparison of the cavity density

function and the value of stress parameters.

It continues in the next page

VI. Discussion 89

Figure 5.4. a) Variation of the von Mises stress (VMS), b) maximum principal stress (MPS),

c) principal stress facet (PFS), and d) stress triaxiality factor ( ) compared with volumetric

cavity density along the notch radius (The stress parameters were obtained from FE

simulation).

tf

The stress parameter showing the best similarity with cavitiy density is the triaxiality

factor, . tf Figure 5.4d) shows the comparison of both distributions. Even though the

distributions do not begin in a similar manner, a similar slope and maximum can be observed

VI. Discussion 90

on the distribution of the stress triaxiality factor. The correlation coefficient of these two

functions is the highest of 0.978. The VMS and the MPS have similar correlation coefficients,

however they are significantly smaller than that of . Both, the VMS and the cavity density

distribution begin with a plateau, which changes into a linear increase at about Z ≈ 200 μm.

The slope at this point, however, is higher for the distribution of real damage. At higher Z the

VMS increases monotonously, while the experimental cavity density reaches a maximum

plateau at Z ≈ 1600 μm, followed by a slight decrease towards the external surface. Assuming

that cavity nucleation is primarily affected by the VMS and cavity growth by the stress

triaxiality the whole damage distribution curve could be explained as a superposition of the

two mechanisms. Most of damage found at the maximum of stress triaxiality is in good

qualitative agreement with the theory describing vacancy diffusion under the action of

hydrostatic stress gradient [

tf

157]. Watanabe [66] concluded also that the crack initiation site

and crack growth path in 9Cr–1Mo–V–Nb (P91) steel welded joints coincided well with the

distribution of the stress triaxiality factor. Comparing experimental creep damage

distributions with computed FE results Li et al. [75] have found that both creep strain

concentration and high stress triaxiality have to be considered to explain creep void formation

and growth; a result similar to that obtained in this work.

In case of intergranular fracture literature data suggest that the cavity density and the

growth rate of grain boundary voids is described either by the maximum principal stress

[70,73,116,158] or by the principal facet stress [135]. Both parameters show a kink at about

1255 μm, which is difficult to recognize on the measured distribution due to experimental

scatter. We note, however, that the PFS model is expected to be valid for uniform grain

structures showing grain boundary sliding [135]. The E911 martensitic steel reveals in

contrast a band like granular structure, when GBS is more difficult. The fairly high correlation

coefficient obtained for the MPS indicates that it has an important effect on the growth of

cavities. Thus, the maximum of the cavity density is well described by (tf Figure 5.4d), the

possible kink position of the cavity density function is observed at the same position of the

kink of the MPS (Figure 5.4b) which according to the FE simulation it remains fairly stable

for over 60% of the creep time. These two parameters also account fairly well for the slope of

the distribution under the multiaxial stress state applied. However, the beginning of the

distribution where the density of cavities is small the VMS, considered to affect the

nucleation, correlates better with the cavity density distribution. Furthermore, the direction of

the MPS changes over 90° along the sample length, however, no preferential orientation of

the voids was found for the azimuthal angle as a function of Z.

VI. Discussion 91

An indirect hint about the cavity growth mechanism is obtained by comparing the

experimental size distribution with the function proposed by Riedel [13], which takes into

account both continuous cavity nucleation and growth. The agreement is remarkable and the

exponents α and β determined from the fit have a high degree of confidence. 2≅β is in

good agreement with the constrained diffusional mechanism [12,13], which predicts a growth

rate for cavity radius proportional to . The second exponent -2ea 1≅α indicates that the size

distribution function depends on time. This non-stationary feature is related to continuous

nucleation, which decreases the average distance between cavities.

VI. Conclusions and perspectives 92

CHAPTER VI

CONCLUSIONS AND PERSPECTIVES

It was shown that in situ microtomography is an excellent method for the investigation

of specimens undergoing inhomogeneous deformation allowing a good characterization of the

kinematics of steady-state creep. The good accuracy of the method is related to the use of

powerful image correlation algorithms, which applied to 3D tomographic images precisely

capture changes in specimen’s shape. The possibility to apply different loads on the sample

like temperature, electric or magnetic fields, shows the flexibility of the experiment, which in

present case was performed in a 5 dimensional parameter space (three spatial dimensions, the

temporal dimension and a temperature gradient).

Activation energies of steady-state creep obtained with the new tomographic method on

small samples are in good agreement with values from literature and those obtained by

conventional methods on large specimens. The good agreement has two reasons: i) the

tomographic method uses the true values of strain-rate and stress and ii) the slabs chosen as

structural unit to characterize the inhomogeneous deformation still contain a large number of

dislocations to fulfill the conditions necessary to treat them on the basis of continuum

mechanics. The newly developed tomographic method should be attractive for the evaluation

of activation energy in expensive materials or in cases when only a limited amount of material

is available.

The method allows also evaluating the apparent activation energy in steady-state creep

as a function of time or strain. Then, any change in the activation energy can be related with

microstructural changes such us precipitation of new phases, coarsening of the already

existing ones or changes in the dislocation density. In this case, however, additional

investigation of the microstructure evolution is needed. With the increase of the tomographic

resolution, the error of can be further reduced, which is mainly determined by the

resolution of the tomographic technique. However, to increase the resolution detectors with

better efficiency at energies above 50 keV are needed.

aQ

From a single in situ experiments lot of information is available, from the growth

evolution of single cavities presented in [159] to damage development as function of

temperature and time or the extraction of material parameters, such as apparent activation

energies of steady-state creep. Therefore, it would be important to plan long-term creep

VI. Conclusions and perspectives 93

experiments in cooperation with synchrotron radiation laboratories in order to study the creep

damage of engineering materials, under conditions characteristic of engineering applications.

In situ experiments with varying stress and constant temperature could be also studied

to evaluate the stress exponent, , in case of a homogeneous material undergoing creep.

Microtomography could be also applied to the study of creep behavior of inhomogenous

materials (i.e. composition or grain size gradients). The possibility to apply electric or

magnetic fields to specimens studied in situ opens up a broad range of experiments delivering

new information about the behavior of engineering materials under complex inhomogeneous

loading conditions.

n

It has been shown that cavities density evaluated from microtomographic

reconstructions is in fair agreement with metallographic results obtained from 2D sections.

Compared to the 2D technique, however, microtomography allows a more detailed

characterization of creep damage. The cavities shape can be well estimated based on the

ellipsoid fit and the shape descriptors defined from it, namely complexity factor (CF ),

elongation ( ) and flatness ( ). From this approximation, statistically relevant information

on the size, shape, orientation and number of cavities in bulk sample could be obtained. Based

on the selection of proper CF and values, the non-coalesced cavity population could be

identified and used for a more straightforward analysis. The study of pore size distributions

suggest that growth of cavities in the E911 steel is governed by the constrained diffusional

mechanism. The study of different deformation states will allow obtaining reliable

information about cavity nucleation, which was very difficult until now.

e f

e

Tomography delivered additionally the continuous distribution of damage along the

notch radius, which could be directly compared to the distribution of stress parameters

obtained from FE modeling. This comparison indicates that cavities density found by

microtomography is most probably affected by the combined action of the equivalent von

Mises stress and stress triaxiality. However, due to its high cross-correlation coefficient the

maximum principal stress can also influence damage development. Although its effect might

be recognized from the location of the kink present in both the MPS and cavity density

distribution, a random spatial orientation of cavities was found in the XY plane.

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VIII. CV 105

Curriculum Vitae

PERSONAL INFORMATION

Last Name SKET

First Name FEDERICO

Nationality Austrian

Date and place of birth 11th of June 1979, Cruz del Eje, Argentina

EDUCATION AND TRAINING

Since March 2006 Max-Planck Institute for Iron research, Düsseldorf

(Germany) Department of Material Diagnostics and Steel Technology, PhD studies on Tomographic characterization of creep damage.

December 2004 to November 2005

Vienna University of Technology, Vienna (Austria), Institute of Materials Science and Technology, Diploma Thesis on Characterization of corrosion resistance of functionally graded hardmetals.

March 1998 to February 2006 National University of Comahue, Neuquén (Argentina), Faculty of Engineering, Electrical Engineering Degree.

tomographic investigation of creep and creep damage under inhomogeneous loading conditions ·...

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