Modeling the interaction between creep deformation and scale growth process

Acta Materialia 55 (2007) 189–201 www.actamat-journals.com

Modeling the interaction between creep deformation and scale growth process Andi M. Limarga, David S. Wilkinson

*

Department of Materials Science and Engineering, McMaster University, 1280 Main Street West, Hamilton, Ont., Canada L8S 4L7 Received 1 May 2006; received in revised form 27 July 2006; accepted 30 July 2006 Available online 11 October 2006

Abstract When an alloy is oxidized (or nitrided) at high temperature with the presence of an applied stress, the scale growth rate is dependent on the magnitude and direction of the stress. Here we present a model to simulate the scale growth process under an applied bending load. This model explores the interaction between creep deformation and the scale growth process, and is able to predict the stress-driven scale growth rate, the creep response of the multilayer and the stress accumulation in the thin scale. A modest applied stress induces large stresses in the scale due to creep rate mismatch between the metal and the scale. This large stress, in turn, alters the diffusion potential governing the scale growth process, leading to a perturbation from a typical parabolic growth rate. The thin scale is also shown to influence the creep response by promoting stress redistribution in the metal–scale system.  2006 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Creep; Oxidation; Residual stresses; Modeling

1. Introduction Many engineering components operating at high temperature are subjected to both an aggressive oxidizing environment and mechanical stress. For example, TiAl is expected to operate at 650–800 C with a working stress of 40–150 MPa in a typical service of turbine blade, automotive valves or turbocharger rotors [1]. Despite the complex service condition to which a component is subjected, most research on high-temperature materials focus solely on oxidation properties or solely on the mechanical behavior (with a particular interest on creep properties) of the material. These studies are performed mainly to improve the properties of high-temperature materials through better creep resistance and the formation of a thin protective oxide scale. Studies on the interaction between mechanical stress and the scale growth process can be classified into three *

Corresponding author. Tel.: +1 905 525 9140; fax: +1 905 528 9295. E-mail address: [email protected] (D.S. Wilkinson).

categories. The first group treats the stresses associated with the oxidation process. These stresses typically include thermal and intrinsic growth stresses. Between these two types of stresses, only the thermal stress is well understood and can be readily analyzed using the theory of elasticity [2]. The source of the intrinsic stress, however, is not so clear, although it has been shown to govern the protectiveness of the oxide scale [3]. A number of studies were performed to investigate the intrinsic stress that develops during oxide scale growth [4–7]. Recently, it was suggested that the compressive growth stress is generated mainly by the formation of the new oxide within the pre-existing oxide scale, a concept similar to that proposed by Rhines and Wolff [8–11]. The deposition of new oxide along the grain boundaries of the pre-existing oxide scale involves lateral expansion that is constrained by the neighboring grains and by the underlying metallic substrate, leading to a large compressive stress in the oxide scale. A recent result by Nychka and Clarke [12] showed that the magnitude of the compressive stress in the oxide scale grown on a thin metallic strip is lower than that on

1359-6454/$30.00  2006 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2006.07.030

190

A.M. Limarga, D.S. Wilkinson / Acta Materialia 55 (2007) 189–201

a thicker metal due to the stress relaxation by creep deformation of the alloy. This study provides an excellent example of how the creep deformation interacts with the oxidation process. The second group of studies investigated the effect of applied stress on oxide scale growth rate on a wide range of materials: nickel [13], nickel–chrome alloy [14] and steel [15]. These oxidation studies were performed under applied tensile load and revealed accelerated oxidation kinetics due to the tensile cracking in the oxide layer. However, recent work on internal nitridation during creep deformation of nickel-based superalloys [16] and on silicon [17] showed that the scale grown on samples under a tensile stress is thicker than that grown without any applied load even without the existence of any tensile crack in the scale. These results are supported by a theoretical analysis by Evans et al. [18] who suggested that tensile stress will accelerate while compressive stress will decelerate the scale growth process by altering the equilibrium vacancy concentration critical to the solid-state diffusional process. However, it was also highlighted in the proposed model that large stresses are required (in the order of several GPa) to influence the scale growth kinetics significantly. It is worth noting that the applied stress in the above studies [13–17] is in the order of 100 MPa, well below what is suggested by this theory. Therefore, there is still an unanswered question of why the application of modest stress can influence the oxidation kinetics so significantly. In the third group of studies, it was shown that the creep properties of an alloy are influenced by the environment in which the creep testing is conducted [19]. Typically, these studies involve a comparison between the creep response obtained in an oxidizing atmosphere such as air and that obtained in an inert gas. The development of a new material as the product of oxidation, which can be in the form of an elastic oxide scale or a soft and non-load-bearing zone (e.g. a continuously spalling oxide scale or the loss of c 0 near the metal–scale interface during oxidation of nickel-based alloys), is considered to influence the creep response of the material [19]. For example, in a study of creep deformation of a pre-oxidized Fe–1Si alloy the creep rate is lower than that of the untreated alloy, which is thought to be due to dislocation pinning by internal oxide particles [20]. Another example of this effect is found in the work of Bolton and McLauchlin [21], which showed that the creep rate of a 9Cr–1Mo steel tested in a CO2 atmosphere is significantly lower than that tested under vacuum. On the contrary, the Cu–Cr composite creeps more slowly when tested under vacuum compared with the specimen tested in air, although the creep stress exponent and the creep activation energy were found to be similar in both environments [22]. The author concluded that the difference in the creep rate is due to the formation of a brittle oxide scale during creep testing in air [22]. This continuously spalling oxide scale leads to the loss of cross-sectional area bearing the load, and hence it virtually increases the stress on the sample [19].

Most of the previous studies done in this area have tackled only one aspect of the interaction (either analyzing the effect of a thin scale on creep deformation or investigating the effect of creep deformation on scale growth rate), and a comprehensive model that deals with the two-way interaction of creep deformation and scale growth process is lacking. In order to address these issues, we present an analysis of the inter-relationship between applied stress, creep deformation and scale growth process. Previously, we presented a one-dimensional (1D) analysis of the scale growth process under uniaxial load [23]. In this contribution, we examine the scale growth process under an applied bending load as this configuration allows direct observation of the effect of the sign of the stress on scale growth rate in a single sample. In particular, we are interested in the TiN scale growth on TiAl under flexure load. This paper seeks to fulfill the following objectives:  to present a general method for predicting the stress in any layer due to the applied stress, creep deformation and the intrinsic growth stress during scale growth process;  to examine the interaction between mechanical stress and the scale growth process: the role of creep deformation on scale growth rate and the role of the scale growth process on creep response of the multilayered system.

2. Modeling of the scale growth process under applied bending load 2.1. Synopsis of the model When mechanical stress is applied to a component at a sufficiently high temperature, the material undergoes creep deformation. Typically, nitride or oxide materials have much higher creep resistance than the metals from which they are formed [24]. In order to maintain the displacement continuity, the scale grown under an applied stress is forced to deform, leading to a large stress in the scale (Fig. 1). The large stress in the scale induced by the creep rate mismatch serves two functions: (i) it influences the scale growth rate and (ii) it affects the creep deformation of the multilayered system (Fig. 2). 2.2. Stress induced by creep mismatch during flexure loading We consider a general case in which a multilayered system is subjected to a bending moment and is allowed to undergo creep deformation. This approach is similar to that proposed by Shen and Suresh [25], who analyzed steady-state creep deformation of a metal–ceramic multilayer subjected to thermal load. Fig. 3 offers a schematic view of the steps involved in modeling the bending of a multilayer strip due to residual stresses and external bending: (i) starting from the stress-free condition; (ii) adding

A.M. Limarga, D.S. Wilkinson / Acta Materialia 55 (2007) 189–201

σapp scale

191

σ scale

hold at high temp.

scale

metal

σ metal

metal

Fig. 1. Schematic diagram showing the generation of stress due to the creep mismatch between the metallic substrate and the scale.

Fig. 2. The interaction between applied stress, creep deformation and scale growth process analyzed in this work.

unconstrained strains due to intrinsic strain; (iii) then adding the constrained strain needed to maintain displacement compatibility; (iv) allowing for bending induced by asymmetric stresses; and (v) due to external bending. This model was originally used to calculate the stresses in multilayered systems subjected to pure bending moment and thermal mismatch [26]. Here we use the same methodology to obtain the solution for a creeping multilayered system subjected to both intrinsic strain and an applied bending moment. In order to ensure the strain compatibility at the interfaces, the total strain e(f, s) in the multilayer is expressed as [26]: eðf; sÞ ¼ e0 ðsÞ þ KðsÞ½f  fðsÞ

ð1Þ

where f = z/L0 is the normalized space dimension along the thickness direction, e0 is the uniform component of the strain (Fig. 1c), K = jL0 is the dimensionless curvature of the beam and f ¼ z=L0 is the normalized bending axis. As the oxide growth is included in this analysis, we scale the time t by the nominal time for diffusion processes so that s ¼ tD=L20 is normalized time, where D is the diffusion coefficient of the moving species responsible for the scale growth process (all non-dimensional parameters used in this study are listed in Table 1). When creep and intrinsic strains are considered, the stress in any layer i can be formulated as: Ri ðf; sÞ ¼ ½eðf; sÞ  ei;creep ðf; sÞ  ei;intrinsic ðf; sÞ; i ¼ 1; 2 . . . n ð2Þ where Ri ¼ ri =Ei is the stress in phase i normalized by its elastic modulus, Ei ¼ ½Ei ð1  m0 Þ=½E0 ð1  mi Þ is the ratio of the elastic modulus of phase i to the reference modulus for a biaxial stress state, ei, creep is the accumulated creep strain and ei, intrinsic is the intrinsic strain due to the scale growth process. In general, the creep strain rate can be expressed as [24]: e_ i;creep ðz; tÞ ¼ f ðr; T ; S; P Þ

ð3Þ

where S represents the state variables describing the microstructural state of the material such as dislocation density and grain size, and P represents the material properties, including lattice parameters, diffusion constants, elastic moduli, etc. The normalized creep strain rate is expressed as E_ ¼ e_ L20 =D. The total creep strain rate in the material can be expressed as the summation of primary and secondary creep strain rates: E_ i;creep ¼ E_ i;primary þ E_ i;secondary

ð4Þ

The elastic strain distribution in the multilayered system can be solved by finding the values of three parameters in Eq. (1): e0, K and f. Considering materials with symmetric creep properties (i.e. the bending axis only depends on the geometry of the strip), these parameters can be obtained from three equilibrium conditions, namely: the resultant force due to the uniform strain component is zero, the resultant force due to the bending strain component is zero and the sum of bending moments is in equilibrium with the applied moment [26]. Once the elastic solution is obtained, an iterative approach is used to calculate the stress and strain distributions in the multilayer by calculating the uni_ At any form strain rate, E_ 0 , and the curvature rate, K. given time, the uniform strain and the curvature can be calculated by: e0 ðs þ DsÞ ¼ e0 ðsÞ þ E_ 0 ðsÞDs _ Kðs þ DsÞ ¼ KðsÞ þ KðsÞDs

ð5aÞ ð5bÞ

It is found that the bending axis f can be calculated easily for any time step without having to calculate df=ds directly. We must now formulate an equation for each of these three conditions of mechanical equilibrium. 2.2.1. The resultant force due to the uniform strain component is zero This can be written simply as: n X Z fiþ1 ðsÞ   Ei ½e0 ðsÞ  ei;intrinsic ðf; sÞ  ei;creep ðf; sÞ df ¼ 0 i¼1

fi ðsÞ

ð6Þ

192

A.M. Limarga, D.S. Wilkinson / Acta Materialia 55 (2007) 189–201

z

a

n i+1 i i-1 1

b

c

z = zn+1 z = zn z = zi+1 z = zi z = zi-1 z = z1

εn,intrinsic εi +1,intrinsic εi,intrinsic εi −1,intrinsic ε1,int rinsic

Table 1 Non-dimensional parameters used in the model Symbol

Normalized parameter

z-position

Z

f ¼ Lz0

z-position of interface

zi

fi ¼ Lzi0

Scale thickness

hscale

gscale ¼ hscale L0

Curvature

j

K = jL0

Time

T

s ¼ t LD2

Strain rate

e_

L E_ ¼ e_ D0

Elastic modulus (biaxial stress state)

Ei

0Þ Ei ¼ EEi0ð1m ð1mi Þ

Stress Diffusion constant

ri D

Ri ¼ Erii D ¼ DND;eff

0

n P

Ei

R fiþ1 ð0Þ

e0 ð0Þ ¼ i¼1 P n

εn = ε 0 εi++1 = ε 0 εi = ε 0 εi−−1 = ε0

fi ð0Þ

2

 ei;intrinsic ðf; 0Þdf ð7aÞ

Ei ½fiþ1 ð0Þ  fi ð0Þ

i¼1 n P

E_ 0 ðsÞ ¼

Ei

i¼1

hR

fiþ1 ðsÞ fi ðsÞ

E_ i;intrinsic ðf; sÞdf þ n P

ε1 = ε0

R fiþ1 ðsÞ fi ðsÞ

i E_ i;creep ðf; sÞdf

Ei ½fiþ1 ðsÞ  fi ðsÞ

i¼1

ð7bÞ

d 2.2.2. The resultant force due to bending strain component is zero In the same manner as above, we can write: n Z fiþ1 X fEi ½KðsÞðf  fðsÞÞdfg ¼ 0 ð8Þ i¼1

Mapplied e

fi

The solution to Eq. (8) leads to the expression for the location of the bending axis f as: n o n P 2 2 Ei ½fiþ1 ðsÞ  ½fi ðsÞ fðsÞ ¼ i¼1 ð9Þ n P 2 Ei ½fiþ1 ðsÞ  fi ðsÞ i¼1

Fig. 3. Schematics showing bending of a multilayer strip due to residual stresses and external bending: (a) stress-free condition; (b) unconstrained strains due to intrinsic strain; (c) constrained strain to maintain displacement compatibility; (d) bending induced by asymmetric stresses and (e) external bending. Adapted from Ref. [26].

where fi = zi/L0 is the normalized z-position of interface between layers (i  1) and (i) as shown in Fig. 1. The formulation of such a time-dependent function is necessary to analyze a scale growth process. Solution of Eq. (6) gives the initial uniform strain (i.e. elastic solution, no creep strain has accumulated) and the uniform strain rate during the creep process as follows:

This shows that f depends only on the elastic moduli and thicknesses of the layers. Thus, it can be used to track the shifting of the bending axis at any given time. It should be noted that the definition of bending axis refers to the z-position where the bending strain is zero and it does not represent the neutral axis, which, according to the bending theory, refers to the line in the cross-section of the beam where normal stress is zero. Once the bending axis has been found, the normalized neutral axis, fneutral can be calculated using Eq. (1): fneutral ðsÞ ¼ fðsÞ  e0 ðsÞ=KðsÞ

ð10Þ

2.2.3. The sum of bending moments is in equilibrium with the applied moment For this condition we can write:

A.M. Limarga, D.S. Wilkinson / Acta Materialia 55 (2007) 189–201

M applied ¼

n Z X

fiþ1 ðsÞ

fi ðsÞ

i¼1

Ei fe0 ðsÞ þ KðsÞ½f  fðsÞ

 ei;intrinsic ðf; sÞ  ei;creep ðf; sÞg½f  fðsÞdf

ð11Þ

Solving this yields the initial curvature and the curvature development rate:

3

n P

parabolic law of oxidation due to the accumulation of stress. The mechanical stress in the scale is considered to influence the scale growth rate by altering the equilibrium anion vacancy concentration, i.e. a tensile stress increases the vacancy concentration while a compressive stress reduces the vacancy concentration. It was postulated that

Ei ½e0 ð0Þ  ei;intrinsic ðf; 0Þf½fiþ1 ð0Þ2  fi ð0Þ2   2fð0Þ½fiþ1 ð0Þ  fi ð0Þg  6M applied

i¼1

Kðs ¼ 0Þ ¼

n P

ð12aÞ 2

2

2

3

3

Ei 6fð0Þ½fiþ1 ð0Þ  fi ð0Þ   6½fð0Þ ½fiþ1 ð0Þ  fi ð0Þ  2½fiþ1 ð0Þ  fi ð0Þ 

i¼1

n P

_ KðsÞ ¼

Ei

hR fiþ1 ðsÞ fi ðsÞ

i¼1

½E_ i;intrinsic ðf; sÞðf  fðsÞÞdf þ

R fiþ1 ðsÞ

R fiþ1 ðsÞ

fi ðsÞ

fi ðsÞ

½E_ i;creep ðf; sÞðf  fðsÞÞdf  hR i f ðsÞ Ei fiiþ1 ðf  fðsÞÞ2 df ðsÞ

In this analysis, we have assumed that the intrinsic strain is constant. Furthermore, e0 and E_ 0 are independent of f so that the curvature rate can be simplified as follows: n P

_ KðsÞ ¼

193

i¼1

fi ðsÞ

½E_ i;creep ðf; sÞðf  fðsÞÞdf  E_ 0

h

i ð12bÞ

to insert fresh oxide/nitride at the metal–scale interface, work of order rDX has to be supplied by the chemical reaction [27]. Thus, the vacancy concentration XV is given by:

ii  fðfiþ1 ðsÞ  fi ðsÞÞ h i n 3 3 P 2 2 2 Ei fiþ1 ðsÞ 3fi ðsÞ  fðsÞðfiþ1 ðsÞ  fi ðsÞ Þ þ fðsÞ ðfiþ1 ðsÞ  fi ðsÞÞ

Ei

hR fiþ1 ðsÞ

½E_ 0 ðf; sÞðf  fðsÞÞdf

fiþ1 ðsÞ2 fi ðsÞ2 2

ð12cÞ

i¼1



 rscale DX kT

Using the results from Eqs. (7a), (9) and (12a), the elastic stress in the multilayer can then be calculated from:

X V ¼ X 0V exp

Ri ðf; 0Þ ¼ Ei ½e0 ðf; 0Þ þ Kð0Þ½f  fð0Þ  ei;intrinsic ðf; 0Þ

where X 0V is the equilibrium vacancy concentration with the absence of stress, k is Boltzmann’s constant and T is the temperature. Eq. (17) implies that the diffusion potential, which also determines the diffusion rate, is changed with the presence of stress. The model predicts that large compressive stress is responsible for the slow growing oxide scale compared with its predicted parabolic law [27]. To a first approximation, DX is associated with the volume change from the metal to the oxide/nitride state: DX = Xscale  Xmetal, where Xmetal and Xscale are the volume of the metal and the scale per metal ion, respectively [27]. The anion vacancy flux from the atmosphere moving from the metal–scale interface towards the scale–gas interface (JV) is influenced by the mechanical stress as follows [27]:    DV rDX JV / C V;surface  C V;interface exp ð18Þ kT dscale

ð13Þ

It was also assumed that, during the creep deformation, planar sections of the multilayer remain plane so that the total strain rate and the stress rate can be written as: _ _ EðsÞ ¼ E_ 0 ðsÞ þ KðsÞ½f  fðsÞ  _ R_ i ðf; sÞ ¼ E fE_ 0 ðf; sÞ þ KðsÞ½f  fðsÞ  E_ i;creep ðf; sÞg i

ð14Þ ð15Þ

Similar to the uniform strain and curvature Eq. (5), the stress in the next time step is updated such that: Ri ðf; s þ DsÞ ¼ Ri ðf; sÞ þ R_ i ðf; sÞDs

ð16Þ

Eqs. (15) and (16) describe the stress in the multilayer at any time during scale growth under an applied load. This stress is then used in a diffusional model to calculate the rate of thickening of the scale. 2.3. Effect of stress on scale growth process In the scale growth process controlled by solid-state diffusion, the growth rate of the scale is determined by the flux of defects responsible for the growth process. Anion vacancies move from the metal–scale interface towards the scale–gas interface while cation vacancies are transported in the opposite direction. Evans et al. [27] have previously proposed a model for the perturbation from the

ð17Þ

Assuming that the vacancy concentration at the scale– metal interface (CV, interface) is much greater than that at the scale–gas interface (CV, surface), Eq. (18) can be simplified to give the stress-dependent scale growth rate:   dhscale Di;eff rDX ¼ exp ð19Þ kT dt hscale

A.M. Limarga, D.S. Wilkinson / Acta Materialia 55 (2007) 189–201

where Di, eff is the diffusion coefficient of species i responsible for the scale growth process within the scale equilibrated at its dissociation partial pressure. When the scale growth process is stress-free, Eq. (19) reduces to the classical parabolic law. The thickness of the metallic substrate decreases as the scale grows. Assuming isotropic behavior, the thickness decrease of the metal, dhmetal, can be calculated as follows: sffiffiffiffiffiffiffiffiffiffiffiffi 3 Xmetal dhmetal ¼ ð20Þ dhscale Xscale Oxide and nitride scales typically grow with a preferred orientation [28], leading to a deviation from the isotropic assumption just mentioned. Thus, the expansion in the directions parallel to the interface will be different from that perpendicular to the interface (parallel to the growth direction). However, the discrepancy from different assumptions proves to be insignificant when the thickness of the scale is still very small compared with that of the metallic substrate [23]. 3. Verification of the model: comparison with finite element analysis

0.0020

4. Numerical calculation and an example based on nitridation of TiAl under flexural loading Mechanical Eqs. (7), (9) and (12)–(16) can now be solved together with diffusional Eqs. (19) and (20) to simulate the scale growth and creep processes under bending load iteratively using MATLAB 6.5 (Mathworks, MA). We are particularly interested in the nitridation process of TiAl under flexure load, for which experimental data are presented elsewhere [29]. Thus, the materials properties used in the calculation refer to TiAl and TiN (Table 2). The creep properties of TiAl are obtained from compression creep testing of TiAl in argon and the diffusivity of

0.0006

0.0014 0.0012 0.0010 -6

τ = 1.50 x 10

0.0008

τ = 1.50 x 10

-5 0.0002 τ = 4.05 x 10

-0.0010

-5 0.0000 τ = 8.10 x 10

-0.0012 -0.0014

-0.0002 -5

normalized z-position, ζ

τ = 4.05 x 10

-0.0004

-0.0016

o

T = 900 C, Σ int = 0 -0.0006

-0.500

-0.0008

-6

elastic

0.0006

-0.0006

-6

τ = 1.50 x 10

0.0004

-5

τ = 4.05 x 10

-0.525

elastic

FE result Current model

elastic

0.0018 0.0016

where Ai is the pre-exponential factor for creep deformation of phase i. Furthermore, the intrinsic growth stress is not taken into account in this verification protocol. Fig. 4 shows the excellent agreement that results in term of stress distribution in the multilayer upon creep deformation calculated by the proposed model and by finite element simulation. The initial linear stress distribution in the metal and in the scale refers to the elastic response of the material to the applied load. At the metal–scale interface, the stress in the scale is significantly larger than that in the metal due to the elastic modulus mismatch. During the creep process, the stress in the metallic substrate is redistributed particularly towards the outer fiber (z-position far from the neutral axis); from R = 6 · 104 in the elastic step to about R = 1 · 104 at s = 8.1 · 105. The stress in the scale increases from R = 6 · 104 to R = 1.9 · 103 during creep. Note that symmetric geometry (same scale thickness on both sides of the crept metal) and symmetric creep behavior for both the thin scale and metallic substrate have been assumed. Thus, the stresses in the scale on both tensile and compressive side have the same magnitude but different signs.

-5

τ = 8.10 x 10

Stress in metal, Σ metal

Stress in scale on tens. side, Σ scale on tens.

Eqs. (7a), (7b), (12a) and (12c) have to be solved numerically as the integration terms contain the position- and time-dependent strain rate, ½E_ i;creep ðf; sÞ. Thus, a finite element simulation was performed using a commercial package ABAQUS 6.5 (ABAQUS, Inc., RI) to provide an estimate on the accuracy of the proposed model and the numerical algorithm employed in this work. The center part of a four-point bending crept specimen is modeled using multi-point constraints to ensure the Bernoulli’s hypothesis assumed in Eq. (14). A pure bending moment is then applied to the metal–scale system and the resulting creep deformation is calculated, assuming a constant thickness, non-growing scale case (g = 0.025, which corresponds to 50 lm scale on both sides of a 2-mm-thick metallic substrate). Steady-state creep behavior is assumed for both the

thin scale and the underlying metal, i.e. the Norton power law creep equation is obeyed: n ð21Þ e_ i;creep ðz; tÞ ¼ sign½ri ðz; tÞAi ri ðz; tÞ i

-0.0018

Σapp = 0.0006

-0.5

-5

τ = 8.10 x 10

0.0

normalized z-position, ζ

0.5

0.500

-0.0020 0.525

Stress in scale on comp. side, Σ scale on comp.

194

normalized z-position, ζ

Fig. 4. Stress redistribution in the scale–metal–scale system during creep deformation as calculated by the current model and by finite element analysis.

A.M. Limarga, D.S. Wilkinson / Acta Materialia 55 (2007) 189–201

TiN

Reference

E (GPa) m A (s1 MPan) N X (m3)

168 0.3 1.28 · 1013 3 1.632 · 1029

[30] [30] [29] [29] [31]

E (GPa) m A (s1 MPan) n X (m3) DN,eff (m2/s)

400 0.25 2.78 · 1012 1 1.908 · 1029 9.0 · 1016

[32] [32] [33,34] [33,34] [35] [29]

nitrogen in TiN scale is calculated from the TiN scale growth without any stress at 900 C in N2 [29]. When applied to this bending setup, the model consists of a three-layer strip: nitride scales on the tensile and compressive sides, with metal in the middle. As the specimen consists practically of only the metallic specimen (before the scale grows), the normalization of the parameters listed in Table 1 uses the metal as the reference, i.e. E0 = Emetal and m0 = mmetal. Furthermore, the normalizing space dimension, L0, refers to the initial thickness of the metal L0 = hmetal(t = 0). 5. Modeling results

0.007

Stress in scale, Σ scale

TiAl

Properties

0.008

η

Σ scale Σmetal

0.0025

0.0008 0.0007 0.0006

0.005

0.005 0.0005

0.004

η =0

0.0004

0.003

0.0125 0.002

0.000 0.00000

b

0.0009

0.006

0.001 0.00002

0.00004 0.00006 Normalized time, τ

0.025

0.0003 0.0025 0.025 0.005 0.0002 0.0125 0.025 0.00008

η=0

0.020

Creep strain

Material

a

Maximum stress in metal, Σ metal

Table 2 Material properties used in the calculation

195

0.0005 0.015

0.010 0.0025 0.005

0.005 0.0125 0.025

0.000 0.00000

0.00002

0.00004

0.00006

0.00008

Normalized time, τ

5.1. Non-growing scale: effect of scale thickness on stress redistribution and creep response during steady-state creep process The stress distribution and the overall creep response of the multilayer are strongly dependent on the ratio of thickness of the scale to that of the metallic substrate. In order to illustrate this effect, we have selected several cases with different thicknesses of the non-growing scale while the thickness of the metal is kept constant. Fig. 5a shows the evolution of the stress in the scale and in the metallic part at the metal–scale interface (both taken at the tensile side of the bending sample), respectively, as the multilayer undergoes creep deformation. Stress relaxation is observed in the metallic part and the load is transferred to the scale. As the mechanical equilibrium suggests, thinner scales are subjected to larger stresses, reaching R = 8 · 103 in the lowest scale thickness to metal ratio considered in this work, g = 0.0025. The stress relaxation process in the metallic part is also facilitated by a thicker scale. The stress drops from R = 8 · 103 to R = 1.4 · 104 after steady-state creep deformation for s = 7.8 · 105 with the thick scale (g = 0.025). In comparison, the stress is reduced to only R = 4.5 · 104 when the thickness ratio is g = 5 · 104. The stress relaxation process has a direct role on the subsequent creep deformation of the multilayer, as can be seen in the accumulated creep strain depicted in Fig. 5b. In this figure, we have also included the reference case

Fig. 5. The effect of scale thickness on (a) stress evolution during creep sequence on stress at the metal–scale interface and on stress at the outer fiber of the metal and (b) on the accumulated creep strain. The calculations were performed for the case of T = 900 C, Rapp = 6.5 · 104.

where a single-layer metal is considered without any scale as the reference. As the scale gets thicker, less creep strain accumulates in the system since the stress in the metal drops much more quickly. Despite the fact that the scale is still significantly thinner than the metallic part, the creep deformation of systems with a thicker scale is much slower than that with a thinner scale. Previous parametric studies revealed that this process is dominated by the creep properties of the metallic substrate, and not by the properties of the scale [23]. This reflects the fact that this study was performed with the assumption that the creep rate of the scale is still several orders of magnitude lower than that of the metal. 5.2. Growing scale 5.2.1. Effect of intrinsic growth stress on scale growth rate Typically, scale growth is accompanied by the generation of a large residual stress due to the intrinsic growth strain [7,9,10]. Evans’s analysis showed that the intrinsic stress could influence the scale growth rate [18]. To evaluate this effect, we have determined the scale growth kinetics

A.M. Limarga, D.S. Wilkinson / Acta Materialia 55 (2007) 189–201

under different intrinsic stresses without any applied stress (Fig. 6). The normalized intrinsic stress considered in this analysis ranges from Rintrinsic = 0 up to Rintrinsic = 0.005, which represents typical values observed during TiN scale growth on TiAl (0 up to 2 GPa). The stress-free scale growth process obeys the parabolic rate law while the existence of a compressive intrinsic stress produces a perturbation from the parabolic trend. The thickness of the scale grown with Rintrinsic = 0.005 differs by 12% from that grown stress-free (g = 0.0108 compared with g = 0.0124 after s = 7.8 · 105). This calculation shows that a large stress is required to alter the scale growth rate significantly. Certainly, the overall applied stress on the material during component service is not of this magnitude. However, the creep mismatch between the metallic substrate and the scale will induce large stress in the scale despite the modest applied stress. 5.2.2. Effect of applied load on scale growth kinetics Three different applied loads have been selected to illustrate the scale growth process under the influence of applied stress. The applied bending moments were calculated to give maximum elastic stresses at the metal of R = ± 3 · 104 and R = ± 6 · 104. As a reference, the scale grown without any applied load is also presented (Fig. 7). The intrinsic stress was set at Rintrinsic = 0, allowing the applied stress to be the only driving force of the stressed-nitridation process. It can be readily seen that increasing applied stress leads to larger perturbations from the growth kinetics found without any applied load. At s = 7.8 · 105, the scale thicknesses with applied stresses of Rapplied = 6 · 104, Rapplied = 0 and Rapplied = +6 · 104 are g = 9.75 · 103, g = 1.16 · 102 and g = 1.39 · 102, respectively. The stresses in the scale on the tensile and compressive sides of the bend metal at s = 7.8 · 105 are R = 6.04 · 103 and R = 6.23 · 103, respectively. The calculation shows that a modest applied stress can

0.012

Σapplied = 0

1 2 3 4 5

Scale thickness, η

0.010 0.008 0.006 1: Σintrinsic = 0

2: Σintrinsic = -0.00125

0.004

3: Σintrinsic = -0.0025 4: Σintrinsic = -0.00375

0.002

5: Σintrinsic = -0.005 0.000 0.00000

0.00002

0.00004

0.00006

0.00008

Normalized time, τ Fig. 6. Calculated shifting of the scale growth kinetics at 900 C due to the intrinsic growth stress without any applied stress.

0.014

Scale thickness, η

196

Σintrinsic = 0

1

0.012

2 3 4

0.010

5

0.008 -4

1: Σapp = 6x10 , tens. side

0.006

2: Σapp = 3x10-4, tens. side 0.004

3: Σapp = 0

0.002

4: Σapp = 3x10 , comp. side

-4

0.000 0.00000

5: Σapp = 6x10-4, comp. side 0.00002

0.00004

0.00006

0.00008

Normalized time, τ Fig. 7. Predicted scale thickness during scale growth process at 900 C under an applied bending load that corresponds to 3 · 104 and 6 · 104 of outer fiber stress.

induce large stresses in the scale due to the creep rate mismatch and, in turn, influence the scale growth rate. 5.3. Interaction among intrinsic growth stress, creepmismatch stress and scale growth process The last step of the analysis is to consider the scale growth process under applied load with the intrinsic growth stress taken into account, which represents the realistic condition during service of a component. We focus on the stress redistribution in the metal–scale system due to the scale growth process, the effect of applied stress on the scale growth kinetics and the effect of thin scales on the creep response. Fig. 8 shows the modeling result of the stress redistribution in the scale–metal–scale system subjected to an applied bending load (corresponding to an elastic stress of R = 6 · 104 at the outer fiber of the metal) considering the growth of the scale. A significant stress transfer from the metal to the scale is observed. At s = 7.8 · 105, the stress in the metal has been reduced from R = 6 · 104 to about R = 7.5 · 105, while the stresses in scales on the tensile and compressive sides have been increased to R = +3.75 · 103 and R = 8.75 · 103, respectively. Note that, in this analysis, an intrinsic growth stress in the scale has been assumed equal to Rintrinsic = 2.5 · 103. Thus, the multilayer is subjected to a non-symmetric stress distribution, as can be seen from the different stress magnitudes in the scale on tensile and compressive sides, even when the creep sequence has not occurred (elastic stress distribution in Fig. 8). The stress magnitude depicted in Fig. 8 has two direct consequences: (i) the large stress in the scale influences the scale growth rate and (ii) the lower stress in the underlying metal results in a lower creep rate. Fig. 9 shows the calculated scale thickness on both sides of the bending model and the scale grown without any applied load. According to this analysis, an applied

0.001

-5

τ = 8.1 x 10

0.000 -0.001 elastic

0.0004

-0.004

0.0002

-5

τ = 8.1 x 10

0.0000

-0.005

-5

-0.006

-5

-0.007

τ = 8.1 x 10

-0.0002 τ = 4.05 x 10

-0.0004

τ = 8.1 x 10

-6

τ = 4.05 x 10

-0.0006

-0.002

-0.003

τ = 8.1 x 10

-5

-0.008

-5

-0.009 -0.500 -0.495 -0.490 -0.485

-0.5

normalized z-position, ζ

0.0

0.485 0.490 0.495 0.500

0.5

Normalized z-position, ζ

normalized z-position, ζ 4

Fig. 8. Stress evolution in the scale–metal–scale system during the growth of the scale under 5.96 · 10 2.5 · 103.

bending load corresponding to a stress at the outer fiber of the metal R = 6 · 104 results in +20% and 14% difference when the thickness of the scales on tensile and compressive sides are compared with that grown on the reference sample. The effect of the thin scale on the creep response of the multilayer is summarized in Fig. 10 by comparing the creep strain rate as a function of creep strain. In this figure, different diffusivity constants (different D* = D/DN, eff ratios with DN, eff listed in Table 2) are selected to illustrate the effect of scale growth rate on the creep response of the multilayer. A higher scale/metal thickness ratio (due to the higher diffusivity constant) leads to a slower creep deformation of the multilayer as the scale is more creep-resistant than the metal. Fig. 10 also reveals a rather interesting observation on the creep rate of the multilayer, particularly at the higher strain. When the metal without any scale undergoes creep deformation under bending load, the creep strain rate always decreases due to the reduction of stress acting in the metal. A thin scale (in the case of D* =

of applied stress considering an intrinsic stress of

Fig. 10. Effect of the scale growth on the creep strain rate of the metal considering different diffusivity (D*) values of the diffusing species.

0.01) is able to reduce the creep strain rate significantly, but the shape of the curve remains unchanged. However, in the cases where thicker scales are considered (D* > 1), 1000

T: Tens. side R: Reference sample C: Comp. side

T R

Scale thickness, η

0.010 C 0.008 0.006 0.004

-3

Σintrinsic = - 3.75x10

Bending sample: Σapp = 6x10

0.002

-4

Reference sample: Σapp = 0 0.000 0.00000

0.00002

0.00004

0.00006

0.00008

Normalized time, τ

Fig. 9. Calculated scale thickness grown with and without an applied load considering an intrinsic growth stress of 3.75 · 103.

0.006

Normalized creep strain rate,

0.012

0.007

100

strain rate, metal strain rate, scale stress, metal stress, scale

0.005 0.004 0.003

10 0.002

Normalized stress, Σ i

0.002

elastic

elastic

0.0006

0.003 τ = 4.05 x 10-5

197

Stress in scale on comp. side, Σscale on comp.

-5

τ = 8.1 x 10

0.004

Stress in metal, Σ metal

Stress in scale on tensile side, Σ scale on tens.

A.M. Limarga, D.S. Wilkinson / Acta Materialia 55 (2007) 189–201

0.001 1 0.000

0.001

0.002

0.003

0.004

0.005

0.006

0.000 0.007

Creep strain

Fig. 11. Comparison between creep strain rates and stresses in the metal and in the scale on the tensile side due to the growth of the scale. The applied stress is 6 · 104.

A.M. Limarga, D.S. Wilkinson / Acta Materialia 55 (2007) 189–201

100

Creep strain rate ratio,

metal

scale

198

10

1

non-growing scale 1: η = 0.0005 2: η = 0.0025 3: η = 0.005 4: η = 0.01 5: η = 0.015 6: growing scale 1E-4

5 4

3 6 2

1E-3

1

produce the similar perturbation from the stress-free scale growth kinetics with a large intrinsic stress (2 GPa). The calculated stresses during creep deformation (at high temperature) for the scales grown under compression, tension and without any applied stress are R = 8 · 103, R = 3 · 103 and R = 2.25 · 103, respectively, for the case of scale grown with an applied stress of Rapplied = ± 6 · 104. Upon cooling to room temperature, additional compressive stress develops due to the thermoelastic properties mismatch between the metal and the scale. This thermal stress in the scale can be estimated according to the beam theory as follows [2]:

0.01

Creep strain in the metal, ε metal

Fig. 12. Ratio between creep strain rates of the metal and the scale as a function of creep strain in the metal for multilayers with different nongrowing scale thicknesses and for the multilayer with a growing scale. The applied stress is 6 · 104.

a significant change of the slope of the curves is observed at the higher creep strain regime, indicating a change in the controlling mechanism as described in Fig. 11. The decrease of the strain rate is dictated by the everdecreasing stress in the metal due to the load transfer to the scale. At low strain, the strain rate of the metal is still at least two orders of magnitude larger than that of the scale (i.e. most of the strain in the scale is still elastic). At a higher strain, however, the stress in the scale becomes very large while the stress in the metal becomes reduced significantly. Thus, the creep rate of the scale becomes comparable with that of the metal (Fig. 12 shows the ratio between the creep strain rate of the metal and the scale for different scale thicknesses). Hence, the creep deformation of the scale controls the overall creep response of the multilayer at the high strain region. 6. Discussion and implications The preceding analysis has revealed the inter-relationship between creep deformation, intrinsic stress and scale growth process. As mentioned earlier, previous analysis by Evans et al. [18,27] showed that a large stress is required to change the scale growth rate significantly. The model proposed in this work reveals that a small applied stress can induce such large stresses due to the creep rate mismatch between the metal and the oxide/nitride scale. In a typical condition, the external mechanical stress is rather small compared with the intrinsic stress (about an order magnitude smaller). However, the modest applied stress has been shown to cause the generation of large stress in the scale due to the creep deformation mismatch between the metal and the thin scale. The stress induced by creep mismatch is in the same order of magnitude with the intrinsic stress so that it can alter the scale growth rate. Thus, only a small applied stress (±100 MPa) is required to

Rscale ¼

amet  ascale DT 1 þ gEscale

ð22Þ

where a is the thermal expansion coefficient and DT is the temperature drop from the nitridation experiment to the room temperature. Taking 9 · 106/C and 12 · 106/C for the thermal expansion coefficients for TiN and TiAl, the thermal stress is estimated to be Rscale, th = 3 · 103. Thus, the approximate residual stresses at room temperature for the scale on the compressive side, on the tensile side and on the neutral specimen are R = 1.1 · 102, R = 0 and R = 5.25 · 103, respectively. Such a large stress can induce mechanical failure of the scale by cracking and/or spallation. Another important finding revealed by the model is that a thin scale can influence the overall creep response of the multilayer (Fig. 5b). Thicker scale promotes faster stress relaxation in the metallic substrate. Since the overall creep deformation of the multilayer depends strongly on the creep rate of the metal, a larger scale/metal thickness ratio results in slower creep deformation of the multilayered system. This is particularly of interest when a thin metal is subjected to an aggressive high-temperature environment in a small-scale application such as a microturbine system. There are several limitations of the model proposed in this study. First, the model has assumed that flat interface and perfect bonding exist between the metal and the scale. When the scale cracks, it loses its load-bearing capability and the stress is relaxed. In such an event, both scale growth kinetics and the creep response of the multilayer will be altered. Better understanding of the crack initiation and propagation processes is required, particularly when dealing with a cyclic oxidation process. The second limitation refers to the assumption that the growth of the scale itself does not influence the properties of either the underlying metal or the thin scale. The growth of subscale phase near the metal–scale interface (e.g. the loss of c 0 particles in oxidation of Ni-based alloys or the growth of aluminum-rich phases in nitridation of TiAl) is not taken into account in the present study. Recently, it has been suggested that the oxidation process involves the injection of vacancies that promotes the dislocation climb process and hence accelerates the creep deformation of the metal [36–38]. The third limitation is related to the

A.M. Limarga, D.S. Wilkinson / Acta Materialia 55 (2007) 189–201

nitridation experiment of TiAl. Microstructural characterization shows that the nitride scale forming on TiAl upon exposure in pure nitrogen environment is a heterogeneous structure consisting of a mixture between TiN and Ti2AlN phases. In this paper, we have treated the scale as a homogeneous layer with uniform properties across the thickness (details of this approach are presented in Ref. [29]). However, incorporating the growth of the subscale and the multilayered scale is not formidable as the proposed model has the capability of including as many layers as necessary, provided that a better description of the growth of the individual layers (i.e. evolution of the concentration profile across the metal and the scale) is available. Despite the simplifications mentioned above, a good agreement is still obtained between modeling and experimental results for a nitridation process of TiAl under an applied bending load. Fig. 13 shows the comparison between the two in terms of nitride scale growth kinetics and the creep response of the sample tested at 900 C in nitrogen. Details of the comparison are offered in Ref. [29].

Nitride scale thickness (μ m)

a

30 tens. side Measured Calculated

25

reference

fitted

20

comp. side

15

5 o

γ -TiAl, σapp = 102MPa, T = 900 C

0 20

40

60

80

100

120

Time (hour)

b

7. Summary We have proposed a model to evaluate the interaction between creep deformation, intrinsic stress and scale growth process. The key findings of this study are as follows: 1. When the scale is able to maintain its adherence to the underlying metal, the large compressive intrinsic stress induced during the growth of the scale is able to reduce the scale growth kinetics. This result is in agreement with the theory proposed by Evans et al. [18]. 2. Modest applied stress (Rapplied  Rintrinsic) has been shown to cause the generation of large stress in the scale due to the creep deformation mismatch between the metal and the thin scale. The stress induced by creep mismatch is in the same order of magnitude as the intrinsic stress so that it can alter the scale growth rate. Thus, only a small applied stress is required to produce the similar perturbation from the stress-free scale growth kinetics with a large intrinsic stress. 3. The model also predicted that the thin scale can influence the creep response of the multilayer by promoting stress redistribution in the multilayer. The load is transferred from the metal to the scale, leading to a large stress in the scale and a small stress in the metal. Thus, the growth of a creep-resistant scale slows the overall creep deformation of the metal–scale system. It is also shown that this effect is amplified when the multilayer possesses a high scale/metal thickness ratio.

Acknowledgements

10

0

199

The authors dedicate this paper to the memory of the late Prof. George C. Weatherly. Discussions with Profs. Gary Purdy, David Embury and Nikolas Provatas are also gratefully acknowledged. The financial support of the Natural Sciences and Engineering Council of Canada and of Pratt and Whitney Canada are gratefully acknowledged. Appendix. Decomposition of total strain into uniform and bending components

Measured Calculated -7

Creep strainrate(/s)

10

In Section 2, the total strain in the multilayer has been decomposed into uniform and bending components, as written in Eq. (1): -8

10

eðf; sÞ ¼ e0 ðsÞ þ KðsÞ½f  fðsÞ 102 MPa σapp = 78 MPa

-9

10

0.000

31 MPa 0.002

T = 900oC, crept in N2 0.004

0.006

0.008

0.010

0.012

0.014

Creep strain

Fig. 13. Comparison between modeling and experimental results for the case of nitridation of TiAl under bending load at 900 C: (a) nitride scale thickness; (b) creep response.

ð1Þ

Thus, the mechanical equilibrium condition of the internal force in the multilayer is given by: n Z fiþ1 ðsÞ X Ei fe0 ðsÞ þ KðsÞ½f  fðsÞ  ei;intrinsic ðf; sÞ i¼1

fi ðsÞ

 ei;creep ðf; sÞgdf ¼ 0

ðA:1Þ

Eq. (A.1) has been decomposed to give two conditions explained in Section 2: (i) internal force due to the uniform

200

A.M. Limarga, D.S. Wilkinson / Acta Materialia 55 (2007) 189–201

Shifting of bending axisfrom centroid of the substrate

0.000

app

=0

app

= 3 x 10

app

= 6 x 10

-4

-0.002

-0.004

-0.006 -4

-0.008

-0.010 0.00000

0.00002

0.00004 0.00006 Normalized time,

0.00008

0.00010

Fig. A1. Shifting of bending axis from the centroid of the metallic substrate during creep deformation.

jected to a bending load. Our analysis on the multilayered TiN–TiAl–TiN, however, shows that the shifting of the bending axis is very small compared with the thickness of TiAl (2 mm), as shown in Fig. A1. After creep deformation for s = 8.1 · 105 with an applied nominal stress of Rapp = 6 · 104, it was found that the shifting of bending axis is very small, i.e. ½ðf=2  fÞ= ðf=2Þ < 1%. The shifting of bending axis in this case has been promoted by the non-symmetric growth of the nitride scales on tensile and compressive sides as we have used symmetric creep properties for both TiAl and TiN. Thus, it is concluded that the decomposition of the total strain proposed earlier by Hsueh et al. [26] can be applied in our study. However, this analysis has to be revisited when materials with asymmetric creep properties are investigated. References

strain is equal to zero; and (ii) internal force due to the bending component is equal to zero: n Z fiþ1 ðsÞ X fEi ½e0 ðsÞ  ei;intrinsic ðf; sÞ  ei;creep ðf; sÞgdf ¼ 0 fi ðsÞ

i¼1

ð6Þ n Z X

fiþ1



 Ei ½KðsÞðf  fðsÞÞdf ¼ 0

ð8Þ

fi

i¼1

We will test the validity of this treatment by analyzing a single layer with a thickness f subjected to a bending load. Eq. (A.1) can then be written as: Z f    E ½e0 þ Kðf  fÞ  ei;intrinsic  ei;creep  df ¼ 0 ðA:2Þ 0

When Eq. (A.2) is decomposed into the uniform and bending components, we can write: Z f Z f Kðf  fÞdf ¼ 0 ðA:3Þ E ½e0  ei;intrinsic  ei;creep df þ E 0

0

It can be shown that the second term in the left-hand side of Eq. (A.3) leads to   Z f f   Kðf  fÞdf ¼ E Kf  f ðA:4Þ E 2 0 Therefore, in order to make the second term of Eq. (A.4) equals to zero, i.e. satisfying the condition ‘internal force due to the bending component is equal to zero’ in Section 2, the following condition should prevail:   f f ¼0 ðA:5Þ 2 Eq. (A.5) implies that the bending axis should remain at the center of the layer. Thus, this treatment is valid when materials with symmetric creep properties are considered. In this case, the bending axis is dictated only by the geometry of the system. On the other hand, the bending axis will be shifted in a material with asymmetric creep properties sub-

[1] Zhang WJ, Deevi SC. Intermetallics 2003;11:177. [2] Timoshenko S. Strength of Materials, Part II: Advanced Theory and Problems. New York, NY, USA: Robert E. Krieger Publishing Co.; 1976. [3] Evans AG, Mumm DR, Hutchinson JW, Meier GH, Pettit FS. Prog Mater Sci 2003;46:505. [4] Pieraggi B, Rapp RA. Acta Metall 1988;38:1281. [5] Srolovitz DJ, Ramanarayanan TA. Oxid Met 1984;22:133. [6] Evans HE. Corros Sci 1983;23:495. [7] Pilling NB, Bedworth RE. J Inst Met 1923;29:529. [8] Clarke DR. Acta Mater 2003;51:1393. [9] Krishnamurthy R, Srolovitz DJ. Acta Mater 2004;52:3761. [10] Limarga AM, Wilkinson DS, Weatherly GC. Scripta Mater 2004;50:1475. [11] Rhines FN, Wolf JS. Metall Trans 1970;1:1701. [12] Nychka JA, Clarke DR. Oxid Met 2005;63:325. [13] Moulin G, Arevalo P, Salleo A. Oxid Met 1996;45:153. [14] Rajendran Pillai S, Sivai Barasi N, Khatak HS, Gnanamoorthy JB. Oxid Met 1998;49:509. [15] Calvarin-Amiri G, Huntz AM, Molins R. Mater High Temp 2001;18:91. [16] Krupp U, Orosz R, Christ H –J, Buschmann U, Wiechert W. Mater Sci Forum 2004;461–464:34. [17] Yen JY, Hwu JG. Appl Phys Lett 2000;76:1834. [18] Evans HE, Norfolk DJ, Swan T. J Electrochem Soc 1978;125:1180. [19] Dyson BF, Osgerby S. Mater Sci Technol 1987;3:545. [20] Rolls R, Shahhosseini MH. In: Guttman V, Merz M, editors. Corrosion and mechanical stress at high temperatures. London: Applied Science; 1981. p. 151. [21] Bolton CJ, McLauchlin IR. In: Guttman V, Merz M, editors. Corrosion and mechanical stress at high temperatures. London: Applied Science; 1981. p. 17. [22] Lee KL. Composites A 2003;34:1235. [23] Limarga AM, Wilkinson DS. Mater Sci Eng A 2006;415:94. [24] Frost HJ, Ashby MF. Deformation Mechanism Maps – The Plasticity and Creep of Metals and Ceramics. NY: Pergamon Press; 1982. [25] Shen YL, Suresh S. Acta Mater 1996;44:1337. [26] Hsueh CH, Lee S, Chuang TJ. J Appl Mech 2003;70:151. [27] Evans HE. Int Mater Rev 1995;40:1. [28] Czerwinski F, Szpunar JA. Mater Sci Forum 1996;204–206:497. [29] Limarga AM, Wilkinson DS. Creep-driven nitride scale growth in c-TiAl, Acta Mater 2006. doi:10.1016/j.actamat.2006.07.024. [30] Liu CT, Stiegler JO, Froes SH. Properties and selection: non-ferrous alloys and special purpose materials, vol. 2. ASM Handbook; 1998. p. 926.

A.M. Limarga, D.S. Wilkinson / Acta Materialia 55 (2007) 189–201 [31] Appel F, Oehring M. In: Leyens C, Peters M, editors. Titanium and titanium alloys. Weinheim, Germany: Wiley; 2003. p. 89. [32] PalDey S, Deevi SC. Mater Sci Eng A 2003;342:58. [33] Crampon J, Duclos R. Acta Metall Mater 1990;5:805. [34] Gogotsi YG, Grathwohl G. J Mater Sci 1993;28:4279.

[35] [36] [37] [38]

201

JCPDS-International Centre for Diffraction Data #38-1420. Andrieu E, Pieraggi B, Gourgues AF. Scripta Mater 1998;39(4/5):597. Gourgues AF, Andrieu E. J Phys IV (France) 1999;9(Pr. 9):297. Dryepondt S, Monceau D, Crabos F, Andrieu E. Acta Mater 2005;53:4199.