Stress analysis and fail-safe design of bilayered tubular supported ceramic membranes

Journal of Membrane Science 453 (2014) 253–262

Contents lists available at ScienceDirect

Journal of Membrane Science journal homepage: www.elsevier.com/locate/memsci

Stress analysis and fail-safe design of bilayered tubular supported ceramic membranes K. Kwok n, H.L. Frandsen, M. Søgaard, P.V. Hendriksen Department of Energy Conversion and Storage, Technical University of Denmark, Frederiksborgvej 399, 4000 Roskilde, Denmark

art ic l e i nf o

a b s t r a c t

Article history: Received 7 July 2013 Received in revised form 23 October 2013 Accepted 10 November 2013 Available online 19 November 2013

Supported ceramic membranes based on mixed ionic and electronic conductors are a promising technology for oxygen separation applications. In addition to chemically induced stress under oxygen activity gradients in the materials, strain mismatch between membrane and support gives rise to considerable stress that may compromise mechanical reliability. This paper presents an analysis of stress generated in tubular supported membranes during operation. Closed-form analytical solutions for stresses due to external pressures, strain gradients, and mismatch in materials properties are derived. Stress distributions in two membrane systems have been analyzed and routes to minimize stress are proposed. For a Ba0:5 Sr0:5 Co0:8 Fe0:2 O3
δ membrane supported on a porous substrate of the same material under pressure-vacuum operation, the optimal configuration in terms of minimizing the risk of fracture by ensuring only compressive stresses in the component is achieved by placing the support on the feed side of the membrane. For a Ce0:9 Gd0:1 O1:95
δ membrane on a MgO support, stress due to thermal strain mismatch is as large as that due to oxygen activity gradient. Tailoring the thermal expansion coefficient of the support is an effective method to alleviate the total stress. Failure criteria for membrane fracture under compression are thereafter presented. It is found that the tolerable flaw size for fracture in compression is in the millimeter range for both membrane systems at operating conditions in the range of practical interest. & 2013 Elsevier B.V. All rights reserved.

Keywords: Oxygen separation Supported membranes Tubular membranes Stress analysis Mechanical failure

1. Introduction Materials exhibiting mixed ionic and electronic conductivity have potential applications in gas separation technologies. For instance, ceramics that conduct both oxide ions and electrons can be used for selective net transport of oxygen through a membrane. The oxygen can be used in oxy-fuel combustion in power plant processes designed for carbon capture and storage [1,2], or used in catalytic membrane reactors to produce syngas [3,4]. The achievable oxygen flux is limited partly by the loss of driving force over the membrane, and therefore an efficient design in terms of optimizing the flux would require the membrane to be as thin as possible. An obvious design is to support a thin membrane on a substrate that provides the necessary stiffness for handling and for carrying loads in case of an absolute pressure difference exists over the membrane. The substrate needs to be porous to allow gas access to the membrane. Tubular and planar units with a bilayer architecture have been proposed, but the former has a number of advantages including ease of manufacturing using extrusion and dimensional stability under temperature and oxygen activity gradients.

n

Corresponding author. Tel.: þ 45 2012 0785. E-mail addresses: [email protected] (K. Kwok), [email protected] (P.V. Hendriksen).

0376-7388/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.memsci.2013.11.020

Stresses build up in both the membrane and the support as the device are exposed to different stress generating loadings during manufacturing and subsequent use. In general, these loadings include externally applied pressure differences, nonuniform temperature distributions, and lattice expansion in the materials under oxygen activity gradients. The various stress generating mechanisms pose material strength requirements that can potentially limit the choice of materials and operating conditions, and place stringent quality requirements on the manufacturing process. It is therefore important to have a general method of analysis to quantify the stresses and assess the effects of different loadings and designs. Stresses due to lattice strains have received particular attention in the literature. It is known that mixed conducting ceramics experience lattice expansion when oxygen is lost at high temperatures. The volume change, referred to as expansion on reduction [5] or chemical expansion [6], can be expressed by a functional dependence on the change in oxygen non-stoichiometry in the material. In all applications, an oxygen activity gradient exists across the thickness of the membrane and chemical stresses are induced. Chemical stresses induced under oxygen activity gradients were studied analytically for solid oxide fuel cells interconnects [5,7] and electrolytes [8]. Yakabe et al. [9] and Yakabe and Yasuda [10] performed numerical studies on chemical stress distributions in planar

254

K. Kwok et al. / Journal of Membrane Science 453 (2014) 253–262

solid oxide fuel cell interconnects during transient and steady states. Coupling of oxygen transport and stress was considered in the modeling of chemical stress by Krishnamurthy and Sheldon [11] for planar and Swaminathan and Qu [12] for tubular geometry. For oxygen separation membranes, Atkinson and Ramos [13] applied a simple defect model to calculate the non-stoichiometry profiles through oxides with either predominant ionic or electronic conductivity and plotted the resulting chemical stresses in planar and tubular membranes subjected to different boundary conditions. Blond and Richet [14] studied numerically thermal and chemical stress distributions in a tubular membrane. In their analysis, temperature and oxygen partial pressure fields were first computed by solving the heat transfer and bulk diffusion equations through the membrane. Stresses corresponding to the computed thermal and chemical strain profiles were then determined using finite element analyses. Zolochevsky et al. [15] investigated effects of surface oxygen exchange kinetics and membrane thickness on chemical stresses in tubular membranes by detailed diffusion modeling incorporated in a finite element solver. Zolochevsky et al. [16] analyzed chemical stresses in hollow fiber membranes under sweep gas, vacuum, and pressure operation modes taking into account both bulk transport and surface kinetics. Prior studies on tubular membranes have been limited to monolithic designs (i.e. components containing one material only) where stresses are primarily due to the presence of chemical strain gradients. In supported membranes, the mismatch in elastic modulus, thermal, and chemical expansion coefficients between the membrane and support materials create additional stresses that are comparable to chemical strain-gradient induced stresses. Depending on the support configuration and operating condition, critical stresses can be located in the weaker porous support even though the chemical strain gradient is most often minimal in the support. To the knowledge of the authors, no investigation of stresses in bilayered tubular supported membranes currently exist. The present work aims to fill this gap. To this end, the present study is directed towards understanding the effect of material mismatch on total stresses arising during operation. Closed-form analytical solutions for stress distributions in tubular supported membranes are derived, taking into account the mismatch in material properties. Solutions for stresses in bilayers and multilayers are well known for planar geometry [17–20], but only plane-stress solutions under a uniform temperature change have been given for circular geometry [21,22]. These solutions cannot be directly applied to bilayered tubular supported membranes where stresses are also induced by chemical strain gradients. The closed-form solutions obtained in this study are advantageous for quick exploration of the design space without the need for numerical tools which are more time-consuming. To illustrate the application of the model for fail-safe design variation, stress distributions in two tubular supported membrane systems are analyzed. The first case is a Ba0:5 Sr0:5 Co0:8 Fe0:2 O3
δ (BSCF) membrane supported on a porous substrate of the same material under pressure-vacuum operation. The second example is a Ce0:9 Gd0:1 O1:95
δ (CGO) membrane supported on porous MgO under sweep-gas operation. Practical routes to minimize stress by operating condition variation, and support material and configuration selection are elucidated. Furthermore, failure of the systems under compressive membrane fracture is analyzed and failure criteria are formulated. In analyzing the two membrane systems, the non-stoichiometry profile is either based on realistic simplifications or adopted from reported modeling studies. The presented model for stresses in bilayered tubular supported membranes can be readily combined with oxygen transport analyses that model bulk diffusion and surface exchange. It is not the aim of the paper to pursue such details, but readers are referred to [23] for oxygen transport modeling of membranes supported by porous substrates.

2. Analysis The cross section of a tubular bilayer is shown in Fig. 1, where 2 concentric annuli are bonded together. The inner and outer radii of the bilayer are designated ri and ro, and the radius at the interface rf. Linear elasticity theory is employed to determine the stress distributions in the bilayer. Because of axi-symmetry, all shear stresses vanish and only the radial, tangential, and axial stresses need to be determined. Since long tubes are considered in actual applications and the stresses remote from the ends are of primary interest, plane-strain conditions are assumed. Stress can generally develop due to absolute pressure differences, non-uniform strains resulting from non-stoichiometry and temperature distributions, and mismatch strains between support and membrane. The elastic properties of each layer are characterized by the elastic modulus E and the Poisson's ratio ν. The inner and outer absolute pressures are defined by Pi and Po. In this work, chemical and thermal strains are assumed to vary with radius only and collectively expressed by the internal strain εi,

εi ðrÞ ¼ γΔδðrÞ þ αΔTðrÞ;

ð1Þ

where γ is the chemical expansion coefficient, Δδ is the change in oxygen non-stoichiometry from the reference state, α is the thermal expansion coefficient, and ΔT is the temperature change from the reference state. To determine the stresses, the bilayer can be regarded as two separate monolithic layers interacting only through the interfacial pressure denoted Pf in Fig. 1. The interfacial pressure Pf obviously depends on the material properties, pressure loads and internal strains of the two layers, and needs to be solved for before the stress distributions in the bilayer can be obtained. In each layer, well-known linear elastic solutions can be directly applied with Pf as the unknown variable. Under a pressure difference, the stresses as given by Lamé [24] are found by satisfying force balance at any r. Since the support layer is loaded by the applied inner pressure Pi and the interfacial pressure Pf, the stresses are expressed by sPrr;s ¼

P f r 2f
P i r 2i

sPθθ;s ¼

r 2f
r 2i




P f r 2f
P i r 2i r 2f
r 2i

ðP f
P i Þr 2f r 2i

þ

ðr 2f
r 2i Þr 2

;

ðP f
P i Þr 2f r 2i ðr 2f
r 2i Þr 2

ð2Þ

;

ð3Þ

membrane ro

rf

support ri

Pi z Pf Po Fig. 1. Cross section of a tubular bilayer.

K. Kwok et al. / Journal of Membrane Science 453 (2014) 253–262

where srr is the radial stress, sθθ is the tangential stress, and r is the radial coordinate. Similarly, the stresses in the membrane layer are given by P o r 2o
P f r 2f ðP o
P f Þr 2o r 2f sPrr;m ¼
2 ; r 2o
r 2f ðr o
r 2f Þr 2 sPθθ;m ¼

P o r 2o
P f r 2f r 2o
r 2f

þ

ðP o
P f Þr 2o r 2f ðr 2o
r 2f Þr 2

ð4Þ

;

ð5Þ

where Po is the prescribed outer pressure. The subscripts s and m denote support and membrane. The superscript P indicates that the stresses are due to pressures acting on the boundaries of the each layer. The stresses due to an internal strain εi are given by Timoshenko and Goodier [25]: ! Z rf Z r 2
r 2i Es 1 r r ε dr
r ε dr ; ð6Þ sεrr;s ¼ i;s i;s 1
νs ðr 2f
r 2i Þr 2 ri r 2 ri sεθθ;s ¼

r 2 þr 2i Es 1
νs ðr 2f
r 2i Þr 2

Z ri

rf

r εi;s dr þ

1 r2

Z ri

r

! r εi;s dr
εi;s ;

! Z r 2
r 2f Z ro Em 1 r r εi;m dr
2 r εi;m dr ; rr;m ¼ 1
νm ðr 2o
r 2f Þr 2 rf r rf

sε

sεθθ;m ¼

! Z r 2 þ r 2f Z ro Em 1 r r ε dr þ r ε dr
ε i;m i;m i;m ; 1
νm ðr 2o
r 2f Þr 2 rf r2 rf

ð7Þ

ð8Þ

ð9Þ

where the superscript ε indicates that the stresses are due to internal strain inside the layers. This stress contribution is zero if the internal strain is constant. The total stresses in each layer are obtained by superimposing the solutions due to pressure difference, Eqs. (2)–(5), onto the stresses due to internal strain, Eqs. (6)–(9),

and

255

εmis is the mismatch strain given by

εmis ¼

2ð1 þ νm Þ r 2o
r 2f

Z

ro rf

r εi;m dr


2ð1 þ νs Þ r 2f
r 2i

Z

rf ri

r εi;s dr:

ð19Þ

The mismatch strain is a measure of the difference in areaaveraged internal strains of the support and membrane. The effect of external pressure and material mismatch to the total stress is carried in sP through Pf. The term sP is non-zero when bilayered membranes or external pressure is considered. The axial stress in each layer is determined from the radial and tangential stresses through szz;s ¼
Es εi;s þ νs ðsrr þ sθθ Þ;

ð20Þ

szz;m ¼
Em εi;m þ νm ðsrr þsθθ Þ:

ð21Þ

The last step of the analysis is to substitute the explicit expression of the interfacial traction Pf into the three principal stresses in the support and membrane layers described by Eqs. (10–(13) and Eqs. (20) and (21). The plane-strain solutions presented thus far are valid when the axial displacement is zero. This condition corresponds to membranes that are axially constrained on both ends and pressures are applied only on the lateral surfaces. An alternative design is to hold the tubular membranes on one end and close the other with an end-cap. In this case, the axial displacement is no longer constrained but needs to be determined from the axial force resulting from the applied pressure difference. The stress distributions for this situation can be found by superimposing the stresses caused by the axial pressure onto the plainstrain solutions already developed. As the additional stress field only involves external loads in the axial direction, only the axial stress will be affected. The radial and tangential stresses are still given by Eqs. (10)–(13). To derive the axial stresses, the end-cap is first assumed to remain plane and constrained to move along the tube axis so that the axial strain of the bilayer is independent of the radius. The axial stresses are written as

srr;s ¼ sPrr;s þ sεrr;s ;

ð10Þ

sθθ;s ¼ sPθθ;s þsεθθ;s ;

ð11Þ

szz;s ¼ Es ðεzz
εi;s Þ þ νs ðsrr;s þ sθθ;s Þ;

ð22Þ

srr;m ¼ sPrr;m þ sεrr;m ;

ð12Þ

szz;m ¼ Em ðεzz
εi;m Þ þ νm ðsrr;m þ sθθ;m Þ;

ð23Þ

sθθ;m ¼ sPθθ;m þ sεθθ;m :

ð13Þ

The interfacial pressure Pf is found by invoking displacement continuity at the interface. Since the displacement is expressed in terms of tangential strain as u ¼ r εθθ , the continuity condition can be equivalently written as

εθθ;s ¼ εθθ;m at r ¼ r f

ð14Þ

The tangential strain under plane strain condition is given by   1
ν2s νs εθθ;s ¼ sθθ;s
srr;s þ ð1 þ νs Þεi;s ; ð15Þ Es 1
νs

εθθ;m ¼

  1
ν2m νm sθθ;m
srr;m þ ð1 þ νm Þεi;m ; Em 1
νm

ð16Þ

By substituting Eqs. (10–(13) into Eqs. (15) and (16) and subsequently into Eq. (14), Pf is determined to be ! 1 1
ν2m 2P o r 2o 1
ν2s 2P i r 2i þ þ ε ð17Þ Pf ¼ mis ; S1 Em r 2o
r 2f Es r 2f
r 2i where S1 is a parameter that depends only on the dimensions and elastic properties of the layers, S1 ¼

2 2 2 2 1
ν2m r o þ r f 1
ν2s r f þ r i ð1 þ νm Þνm ð1 þ νs Þνs þ þ
; 2 2 2 Em r o
r f Es r f
r 2i Em Es

ð18Þ

where εzz is the yet unknown axial strain of the bilayer. The axial strain is obtained from satisfying force balance in the axial direction: Z rf Z ro 2π rszz;s dr þ 2π rszz;m dr ¼ P o π r 2o
P i π r 2i : ð24Þ ri

rf

Other end constraints can be handled in a similar manner by replacing the right-hand side of Eq. (24) with the corresponding axial force. Upon substituting Eqs. (22) and (23) into Eq. (24), the axial strain is determined to be 1 εzz ¼ ð1
2νm ÞP o r 2o
ð1
2νs ÞP i r 2i þ2ðνm
νs ÞP f r 2f S2 ! Z Z þ 2Em

ro

rf

r εi dr þ 2Es

rf

ri

r εi dr ;

ð25Þ

where S2 ¼ Em ðr 2o
r 2f Þ þEs ðr 2f
r 2i Þ:

ð26Þ

For membranes tubes that are closed on one end, the stress distributions are found by first evaluating the interfacial pressure, Eq. (17), and the axial strain, Eq. (25). The stresses in the three principal directions are then calculated from Eqs. (10–(13) and Eqs. (22) and (23). Note that the solutions apply to any integrable functional radius dependence of εi.

K. Kwok et al. / Journal of Membrane Science 453 (2014) 253–262

3. Case studies 3.1. Dense BSCF supported on porous BSCF Among various mixed ionic electronic conducting materials, BSCF has been found to possess superior oxygen permeation characteristics at elevated temperatures [26–28]. BSCF membranes deposited on supports of the same material operating are being developed [29,30] and considered for industrial scale oxygen production in oxy-fuel power plants under a pressure-vacuum mode in the European Union Consortium project HETMOC. The choice of support material is based on minimizing mismatch with the membrane. In future industrial scale oxygen production, a low cost and strong support should be applied. A support made of the same material as the membrane is an option in an early development stage. In the base case design, we consider a 50 μm membrane on a support with a thickness of 1.0 mm and an inner radius of 4.5 mm operating at a uniform temperature of 850 1C. The driving force for oxygen permeation is provided by raising the air pressure on the outside while drawing a vacuum inside the membrane. The outer absolute pressure is set at Po ¼10 bar, which gives an oxygen partial pressure of qo ¼2 bar. Vacuum is drawn on the inside such that an inner pressure of Pi ¼0.2 bar is maintained. With pure oxygen inside the membrane, the oxygen partial pressure takes the same value as the absolute pressure, qi ¼0.2 bar. The reference state oxygen partial pressure is 0.2 bar. A semi-empirical equation suggested by Yang and Lin [31] was used by Kriegel et al. [32] to describe experimental data relating oxygen partial pressure q and oxygen non-stoichiometry δ of BSCF. The relation is written as

δ¼

3Kq
n : 1 þ Kq
n

ð27Þ

The parameters K and n take the values of K ¼0.2281 barn and n ¼0.0215 at 850 1C. The oxygen non-stoichiometry values on the inner and outer surface were calculated from the prescribed oxygen partial pressure to be δi ¼0.577 and δo ¼ 0.554 using Eq. (27). The reference oxygen non-stoichiometry is δr ¼0.577. It is noted that the change in oxygen non-stoichiometry from reference to operating condition is zero on the inner surface and negative on the outer surface (oxygen loss is counted positive). In the general case, the oxygen non-stoichiometry profile through the thickness of the support and membrane is determined by considering the oxygen activity through the bulk and on the surfaces of the membrane [3]. In this example, a simplified variation is adopted. The oxygen non-stoichiometry varies with radius as

δ i
δr for r i rr rr f Δδ ¼ ðδ
δ Þ r
rf for r r r r r o o i ro
rf f

The chemical expansion coefficient is taken to be γ ¼ 0.012 [32]. The mechanical properties of ceramics are dependent on porosity in general [33]. The elastic moduli adopted here are Em ¼ 63 GPa [34] and Es ¼33 GPa [35] for dense and porous BSCF, corresponding to porosities of 4.5% and 38% respectively. The Poisson's ratio was assumed to be ν ¼0.33 for both materials. Since the same material is used for the membrane and the support and a uniform temperature change is assumed, no thermal stresses will be present. For stress evaluation, the internal strain can be represented by the chemical strain alone,

εi ¼ γΔδ;

ð29Þ

which is plotted in Fig. 2. The support experiences no chemical strain because the change in oxygen non-stoichiometry is zero. The membrane contracts as the oxygen non-stoichiometry decreases. With the given internal strain profile, the four different integrals appearing in Eqs. (10–(13), (17), (25), (22), and (23) can be readily evaluated to give the stress distributions. The stress distributions are depicted in Fig. 3 for a tubular supported membrane closed on one end. The radial stress is compressive and insignificant when compared to the tangential and axial stresses. Both the tangential and axial components are drastically different in the support and the membrane, and characterized by a discontinuity at the interface. In both cases, the stresses are mild and compressive in the support, but gradually increase from compressive at the interface to tensile at the outer membrane surface, where the highest stresses are located. For the case analyzed, the axial stress is the most critical. The stresses generated by the chemical strain gradient are almost identical in the axial and tangential directions, but the applied pressure difference exerts a higher compressive stress in the tangential direction, which makes the axial stress the largest tensile stress component. It is recalled that the total stress can always be decomposed into a contribution representing the external pressure and mismatch strain, sP , and a contribution representing the strain gradient, sε . The individual contributions are illustrated for the tangential stress component in Fig. 4. As shown in Fig. 4(a), sεθθ;s ¼ 0 because there is no chemical strain gradient in the support. The support stress is entirely due to external pressure and mismatch strain coming from the difference in chemical contraction between the support and membrane. In the

membrane support 0.01 0.005

ð28Þ

A constant stoichiometry is assumed for the support as the oxygen exchange kinetics of porous BSCF substrate is fast. This causes the oxygen exchange to take place very close to the interface between the membrane and support. In this study, a linear variation in oxygen non-stoichiometry is a close approximation for BSCF at the steady state. In fact, a recent modeling study on BSCF predicted a closely linear profile through the membrane in a similar nonstoichiometry range [16]. Surface losses are neglected in writing Eq. (28). The approach used in the present study essentially corresponds to the worst case scenario in terms of stress generation because the non-stoichiometry difference is the greatest with perfect surface reactions. Hence design criteria derived under this assumption will be fail-safe towards the more realistic situation when surface losses are included.

0 −0.005 εi [%]

256

−0.01

r = ri

r = rf

qi = 0.2 bar

qf = 0.2 bar

i

= 0.577

f

= 0.577

−0.015 r = ro

−0.02

qo = 2.0 bar −0.025 o

−0.03 4.4

4.6

4.8

5 r [mm]

= 0.554 5.2

5.4

5.6

Fig. 2. Internal strain across the thickness of BSCF support and membrane at T ¼850 1C. The conditions at the reference state are: qr ¼ 0.2 bar, δr ¼0.5765.

K. Kwok et al. / Journal of Membrane Science 453 (2014) 253–262

25 radial tangential axial

20

Pi = 0.2 bar

Po = 10.0 bar Δqo = 1.8 bar Δ o = -0.023

20 σP 15

θθ,s

10 Pi = 0.2 bar Δqi = 0.0 Δ i = 0.0

0

5 0 −5

−5 −10 4.4

θθ,s

σε

10 σθθ [MPa]

σ [MPa]

15

5

257

−10

4.6

4.8

5 5.2 r [mm]

5.4

5.6

5.8 −15 4.5

5

5.5

r [mm]

25 20

interface support

Po = 10.0 bar

membrane

20 P

10 5

15 10

0 −5 −10 5.4

σθθ,m σε

θθ,m

radial tangential axial

σθθ [MPa]

σ [MPa]

15

σ

θθ,m

5 0 −5

5.45

5.5 r [mm]

5.55

5.6

Fig. 3. Stress distributions in porous BSCF support and dense BSCF membrane at T ¼ 850 1C: (a) across the whole thickness and (b) detailed view of the membrane.

−10 −15 5.5

5.51

5.52

5.53

5.54

5.55

r [mm] Fig. 4. Contributions to total stress by sPθθ and sεθθ in (a) support, and (b) membrane.

membrane, the chemical contraction is self-constrained due to the presence of a gradient, which gives rise to a stress sεθθ;m that is symmetric about the middle of the membrane as shown in Fig. 4 (b). The magnitude of this stress is determined by the chemical strain and the elastic modulus of the membrane material, therefore it can only be reduced by lowering the membrane chemical expansion coefficient or elastic modulus for a given oxygen nonstoichiometry gradient. This is not always possible because the choice of membrane material primarily depends on permeation requirements. A more effective way to reduce the total stress is to alter sPθθ;m . For the present case, the support is not subject to any chemical contraction and so it acts as an additional constraint to the contracting membrane. This gives rise to mismatch strain that induces tensile stresses in the membrane. The external pressures create compressive stresses, but the magnitudes are smaller. The resulting sPθθ;m is therefore tensile. The sign and magnitude of sPθθ;m can be adjusted by careful choice of support configuration and operating conditions. Several design options are explored in the following. The constraint posed on the membrane by the support will instead create compressive stresses in the membrane if the support is subjected to a chemical contraction higher than that in the membrane. This can be achieved by reversing the direction

of the applied pressure difference (and hence the oxygen nonstoichiometry gradient) so that the entire support contracts chemically. In Fig. 5, the tangential stress for Pi ¼10 bar and Po ¼0.2 bar is shown. It is clear from Fig. 5(b) that sPθθ;m has turned compressive, which helps to bring down the overall tensile stress in the membrane. By comparing Figs. 4(b) and 5(b), it is observed that the maximum total stress has reduced to about half. However, the applied pressures induce tensile stress when a higher pressure is maintained inside the supported membrane. The support is put in tension as a result, and becomes the critical failure component in this case. This configuration is clearly not desirable because the porous substrate has a significantly lower fracture strength. The optimal configuration that avoids any appearance of tensile stress is obtained by placing the membrane on the inner surface of the support while maintaining a higher pressure on the outer surface of the support. The tangential stress distribution for this configuration is shown in Fig. 6. The component sPθθ;m becomes even more compressive in Fig. 6(a) than in Fig. 5(b). This is because compressive stresses are created by both the applied pressure and the support which contracts chemically more than the membrane. For membranes deposited on supports made of the same material, the risk of fracture under tensile stress is minimized by positioning the support on the feed side of the membrane. This is

258

K. Kwok et al. / Journal of Membrane Science 453 (2014) 253–262

Pi = 0.2 bar

10

0

5

−0.005

0

−0.01

−5

−0.015

−15

P θθ,s σεθθ,s

σ

σP

εi

θθ,m ε σθθ,m

10

0 −0.005

σθθ,m

0

−0.01 −10 −0.015 −20

εi [%]

0.005

σθθ [MPa]

15

−10

0.005

20

εi [%]

σθθ [MPa]

Pi = 10.0 bar

−0.02

−0.02 εi

−30

−0.025

−0.025

−20 4.5

−40 4.45

−0.03 5.5

5

4.46

4.47

4.48

−0.03 4.5

4.49

r [mm]

r [mm]

Po = 10.0 bar

Po = 0.2 bar

0.005

20

15

0.005

10

0

0

10

−0.01

−5

−0.015 σP

−10

σ −20 5.5

θθ,m

5.51

5.52

5.53

−0.015 σP θθ,m σεθθ,m

−20

−0.02

θθ,m σεθθ,m

−15

−0.01 −10

−30

εi [%]

0

0 σθθ [MPa]

−0.005 εi [%]

σθθ [MPa]

−0.005 5

−0.02 εi −0.025

−0.025 εi 5.54

−0.03 5.55

−40 4.5

r [mm]

r [mm]

because the support will exhibit higher chemical contraction or lower expansion than the membrane in such a configuration, and put compression on the membrane. In the optimal configuration, the maximum compressive stress is located at the inner surface of the membrane. The magnitude of this stress can be lowered by varying operating conditions. An example of such design variations is shown in Fig. 7, where the maximum stress in the membrane is plotted as a function of outer pressure while other parameters are kept constant. The maximum compressive stress decreases quickly with outer pressure. The contribution due to the chemical strain gradient within the membrane decreases only slightly with outer pressure. The overall reduction is mainly due to the drop in the applied pressure and mismatch strain. 3.2. Dense CGO supported on porous MgO Materials based on CGO have been considered as membranes for syngas production [36] because of high ionic conductivity at relatively low temperatures and stability at low partial oxygen pressures of CGO [37]. To support CGO membranes, porous MgO has been proposed as the substrate material due to its low cost and mechanical robustness at high temperatures [38]. The CGO membrane and MgO support in this case study has a thickness of 30 μm and 1.0 mm respectively. The oxygen activity through the CGO membrane is taken from a detailed modeling study by Chatzichristodoulou et al. [23]. After reaching a

Fig. 6. Tangential stress for a membrane deposited on the inner surface of the support: (a) membrane and (b) support. The operating pressures are Pi ¼ 0.2 bar and Po ¼10 bar.

0

σP

θθ,m ε ,m

−5

σθθ

σθθ,m

−10 σθθ (r= ri ) [MPa]

Fig. 5. Tangential stress distribution in the bilayer after reversing direction of overpressure (Pi ¼10 bar and Po ¼0.2 bar): (a) support and (b) membrane.

−0.03 5.5

5

−15 −20 −25 −30 −35 −40

6

7

8

9

10

11

12

Po [bar] Fig. 7. Maximum stress variation with outer pressure for a membrane deposited on the inner support surface. The inner pressure is kept at Pi ¼0.2 bar.

temperature of 800 1C, H2 with 1% H2O is introduced as a sweep gas inside the supported membrane with a flow rate of 104 ml/min. Both sides are maintained at atmospheric pressure, i.e., Po ¼ Pi ¼1 bar. At the steady state, the oxygen partial pressures on the feed and permeate sides are qo ¼0.21 bar and qi ¼5.64  10
18 bar respectively. The reference state oxygen partial pressure is 0.21 bar.

K. Kwok et al. / Journal of Membrane Science 453 (2014) 253–262

The numerical values of the computed oxygen nonstoichiometry profile at T ¼800 1C were directly taken from Chatzichristodoulou et al. [23] and fitted with a cubic polynomial to obtain an integrable form for Δδ. The reference nonstoichiometry is δr ¼0.0. It is assumed that there is no chemically induced strain in the MgO support. The oxygen non-stoichiometry change is summarized as 3

2

c1 ðr
r f Þ þ c2 ðr
r f Þ þ c3 ðr
r f Þ þ c4

for r f r r r r o

−0.48 −0.50

ð30Þ

where the constants c1, c2, c3, and c4 are given in Table 1. A uniform temperature change is considered for both layers. Assuming the CGO membrane and MgO support to be stress-free at a sintering temperature of 1200 1C, the change from the sintering to the application temperature is ΔT ¼
400 1C. For CGO, a chemical expansion coefficient of γm ¼0.1 [23] and a thermal expansion coefficient of αm ¼12.0  10
6 1C
1 [39] were used. The elastic modulus and the Poisson's ratio are taken to be Em ¼ 200.0 GPa and νm ¼ 0.33 respectively [40]. The thermal expansion coefficient of MgO is αs ¼13.9  10
6 1C
1 [41]. The elastic modulus was measured to be Es ¼50.0 GPa and a Poisson's ratio of νs ¼0.23 was used [42]. In this example, stresses arise due to the chemical strain gradient and mismatch in thermal and chemical strain between the membrane and support. The internal strain is the summation of chemical and thermal strain:

α s ΔT for r i r r r r f εi ¼ γ Δδ þ α ΔT for r rr r r m o f m

−0.46

εi [%]

for r i r r r r f

0

membrane support

−0.56 −0.58 4.4

Table 1 Parameters for calculating chemical and thermal strains in CGO/MgO membrane system. Parameter

Value

c1 (mm
3 ) c2 (mm
2 ) c3 (mm
1 ) c4 ΔT (1C)

59.989 5.0829
0.60214 0.011870
400

4.6

4.8

5

5.2

5.4

5.6

r [mm] membrane support 0.14

ð31Þ

The internal strain profile is depicted in Fig. 8, where a chemical expansion is induced by the oxygen activity gradient and a thermal contraction results from the temperature drop from the stress-free state. The stress distributions corresponding to the internal strains in Fig. 8 are plotted in Fig. 9. The support is in slight tension in all three principal stress directions. In the membrane, the tangential and axial components are compressive while the radial component remains mildly tensile. The highest compressive stresses occur at the interface and gradually decrease towards the outer membrane surface. The tangential stress in the membrane is again interpreted with the two stress contributions sPθθ;m and sεθθ;m in Fig. 10. Since no pressures are applied in this case, sPθθ;m is solely due to mismatch strain, which is a result of the difference in both chemical and thermal strains in the membrane and support. The higher thermal expansion coefficient of MgO leads to higher thermal contraction in the support. Since chemical expansion is larger in the membrane at the same time, the two effects add up and lead to a high compressive sPθθ;m . It is clear from Fig. 10 that the stress contribution due to mismatch strain is as significant as the strain gradient contribution. Effective stress reduction requires controlling both the non-stoichiometry gradient and material mismatch. Since MgO is not responsive to oxygen activity, it makes no difference to deposit the membrane on either side of the support. However, the maximum stress can be reduced by tailoring the

−0.52 −0.54

0.12 0.1 qf = 5.64 x 10 -18 bar

0.08 εi [%]

Δδ ¼

259

f

= 0.0119

0.06 0.04 qo = 0.21 bar

0.02

o

= 0.0

0 −0.02 4.4

4.6

4.8

5

5.2

5.4

5.6

r [mm] Fig. 8. Internal strain profile in CGO/MgO supported membrane: (a) thermal contraction under a uniform temperature change of ΔT ¼
400 1C and (b) chemical expansion under the imposed oxygen non-stoichiometry.

support properties. The most effective route is to bring the thermal expansion coefficient of the MgO support to be closer to that of CGO, such that the chemical expansion in CGO can be compensated by thermal contraction and the overall mismatch strain can be reduced. Lowering thermal expansion coefficient of MgO has been demonstrated by doping with yttria-stabilized-zirconia (YSZ) [43]. A thermal expansion coefficient of 12.3  10
6 1C
1 was obtained for YSZ-MgO composites. Stress can also be minimized by using a support with a low elastic modulus. The amount of stress reduction achieved by reducing support elastic modulus is illustrated in Fig. 11. The maximum compressive stress is reduced to about
400 MPa by using a more compliant support with a modulus of Es ¼10 GPa. 3.3. Critical compressive stress In both BSCF/BSCF and CGO/MgO systems, the membranes are under compression in both axial and tangential directions, while the supports are only subject to mild stresses due to the difference

260

K. Kwok et al. / Journal of Membrane Science 453 (2014) 253–262

−350

15 −400 σzz (r=rf ) [MPa]

σ [MPa]

10

5

−450

−500

0

radial tangential axial

−5 4.5

5 r [mm]

−550 10

20

30

40

50

Es [GPa]

5.5

Fig. 11. Reduction of maximum compressive stress by lowering elastic modulus of the support material.

σ1

0

σ [MPa]

−100 −200 −300

σ2 −400

5.5

5.505

5.51

5.515 r [mm]

5.52

5.525

wing crack 5.53

σ1

Fig. 9. Stress distributions in (a) MgO support and (b) CGO membrane.

200

−0.36

100

−0.38

−0.42

−200

−0.44 σP

−300

−0.46

θθ,m ε σ θθ,m

−400

−0.48

σ

εi

θθ,m

5.505

5.51

5.515

εi [%]

σθθ [MPa]

−100

−500 5.5

Fig. 12. Schematic of wing crack initiation from an inclined crack under compression.

−0.4

0

5.52

5.525

σ2 2a

radial tangential axial

−500

initial crack

−0.5 5.53

r [mm] Fig. 10. Contributions to total tangential stress by sPθθ and sεθθ in CGO membrane.

in elastic modulus and thickness between the layers. Under such a stress state, the prominent failure modes are compressive membrane fracture and buckling-driven interface delamination. Hendriksen et al. [44] analyzed buckling-driven delamination in tubular support membranes based on the fracture mechanics analysis established for multilayers [45,17]. Here the emphasis is on membrane fracture under compression. An approximate

analysis is carried out to derive a critical stress criterion for both BSCF and CGO membranes. Brittle ceramics fail in compression by propagation of cracks. Unlike tensile fracture in which a single crack grows unstably (the crack propagates across the sample to cause failure once initiated), crack growth in compression is stable. Under increasing compressive stress, a population of small cracks extend stably in length, and finally interact to cause sample failure. This is the reason why compressive strength is typically much higher than tensile strength in ceramics. One mechanism of compressive failure is the initiation of wing cracks from the tips of initial flaws inclined to the direction of largest compressive stress [46,47]. A schematic of wing crack growth from a single initial flaw is shown in Fig. 12, where s1 is the most compressive stress in the body while s2 is the stress perpendicular to s1. It can be shown that an initial flaw oriented at 451 is the most critical, assuming the crack faces are frictionless. The wing crack initiation condition is given by [47] pffiffiffi
3K IC ; sc ¼ pffiffiffiffiffiffi π að1
λÞ

ð32Þ

where sc is the critical value for s1, KIC is mode I fracture toughness, a is the initial crack size, and λ is the stress ratio s2 =s1 .

K. Kwok et al. / Journal of Membrane Science 453 (2014) 253–262

To apply the crack initiation condition, the membranes are assumed to be in biaxial stress state. This is a close approximation considering the small radial stress compared to the tangential and pffiffiffiffiffi axial components. A fracture toughness of 1:2 MPa m is used for pffiffiffiffiffi BSCF [48] and 1:8 MPa m for CGO [49]. The critical stress is plotted against the stress ratio for several values of initial flaw sizes in Fig. 13. The magnitude of critical stress increases with the stress ratio. This is because the lateral stress works to close the crack and impedes wing crack initiation. Stresses in membranes arising from applied pressure difference and chemical or thermal mismatch always have the same sign and similar magnitude in the tangential and axial directions (the stresses are equibiaxial in planar membranes). This is a desirable characteristic as far as compressive failure is concerned. The stress states for the analyzed cases of BSCF and CGO membranes are marked by crosses in Fig. 13. For the BSCF membrane, the stress values in the optimal configuration with the membrane deposited on the inside of the support are plotted, where s1 and s2 represent the tangential and axial components at the inner membrane surface respectively. For the CGO membrane, the axial stress at the interface is the most compressive, and therefore represents s1 in applying the crack initiation condition. It is recognized that the BSCF membrane is fail-safe in compression because a crack size of larger than 10 mm is required to initiate wing cracks. For CGO, the critical flaw size is slightly less

0 a = 10 mm −50

σc [MPa]

a = 1 mm −100

−150

−200

−250

a = 0.1 mm

0

0.2

0.4

0.6

0.8

1

λ 0 a = 1 mm

σc [MPa]

−200

a = 10 mm

a = 0.1 mm

−400

−600

−800

−1000

0

0.2

0.4

0.6

0.8

1

λ Fig. 13. Critical stress for initiation of wing cracks under compression for (a) BSCF and (b) CGO membranes. The cross indicates the membrane stress state in the present analysis.

261

than 1 mm. Failure in compression does not seem to pose stringent requirements for the membrane systems considered. However, the criteria to avoid buckling-driven delamination should also be considered.

4. Conclusion Explicit closed-formed analytical solutions are derived for stresses in a tubular supported membranes due to external pressures, radial strain gradients and mismatches in material properties between the two layers. The simple formulas are useful tools for evaluating stress distributions when designing tubular supported membranes with different materials, dimensions, and operating conditions. The analytical models have been employed to study two examples of tubular supported membranes relevant for oxy-fuel combustion and syngas production purposes. It has been demonstrated that the radial stress in tubular membranes is in general insignificant. The tangential and axial components are the critical stresses to consider for mechanical reliability. The stresses in the membrane are always composed of a strain gradient contribution due to lattice expansion upon reduction and a contribution due to mismatch strain and external pressure. The strain gradient contribution depends only on the imposed gradients and properties in the membrane. Routes to reduce this contribution are more restricted because they involve lowering the membrane elastic modulus or the driving force. The mismatch strain and external pressure contribution, however, can be effectively varied by careful choice of support material, support configuration, and operating conditions. For a dense BSCF membrane supported on porous BSCF substrate under a pressure-vacuum operation, it is found that the optimal configuration is to place the membrane on the inner surface of the support and pressurize on the outer surface of the support. The stresses resulting are all compressive, which are desirable for the brittle membrane and support materials. In general, for membranes supported on substrates of the same material, the optimal configuration is obtained by placing the support on the feed side of the membrane. Further reduction of stress magnitude can be achieved by lowering the outer pressure. For a dense CGO membrane deposited on a porous MgO support under a sweep-gas operation, the mismatch strain due to chemical and thermal expansion coefficient mismatch is as significant as chemical strain gradient in terms of stress generation. The position of the membrane relative to the support makes no difference to the sign or magnitude of the stress. Tailoring the thermal expansion coefficient of MgO can alleviate the compressive stress magnitude in the membrane. The membrane stress can also be reduced by choosing a support with lower elastic modulus. In both cases, the membrane is put under biaxial compression. An analysis of compressive membrane fracture reveals that the tolerable flaw size is in the millimeter range and does not appear to be a stringent requirement. The present analysis is limited to radial strain gradients and linear elastic behavior. Strain gradients along the axis of the tube can also exist as a result of, for instance, the transport of oxygen gas from the membrane surface to the connection point of the tube. Axial gradient is neglected in this study because it can be made sufficiently small by the design of a very open porous support structure. Creep is also expected as ceramic membranes operate at high temperatures. In the presence of creep, the tensile stresses due to constrained chemical contraction will be relaxed over time. Future studies should therefore consider creep deformation.

262

K. Kwok et al. / Journal of Membrane Science 453 (2014) 253–262

Acknowledgments The research leading to the findings presented in this work is funded by the European Commission under the 7th Framework Programme (FP7/2007-2013), grant agreement no. 268165 (HETMOC). References [1] R. Bredesen, K. Jordal, O. Bolland, High-temperature membranes in power generation with CO2 capture, Chem. Eng. Process. 43 (2004) 1129–1158. [2] H. Stadler, F. Beggel, M. Habermehl, B. Persigehl, R. Kneer, M. Modigell, et al., Oxyfuel coal combustion by efficient integration of oxygen transport membranes, Int. J. Greenh. Gas Control 5 (2011) 7–15. [3] P. Hendriksen, P. Larsen, M. Mogensen, F. Poulsen, K. Wiik, Prospects and problems of dense oxygen permeable membranes, Catal. Today 56 (2000) 283–295. [4] H. Bouwmeester, Dense ceramic membranes for methane conversion, Catal. Today 82 (2003) 141–150. [5] P. Hendriksen, J. Carter, M. Mogensen, Dimensional instability of doped lanthanum chromites in an oxygen pressure gradient, in: M. Dokiya, O. Yamamoto, H. Tagawa, S. Singhal (Eds.), Proceedings of the Fourth International Symposium on SOFC, Yokohama, 18–23 June 1995, vol. 95-1, The Electrochemical Society, 1995, pp. 934–943. [6] S.B. Adler, Chemical expansivity of electrochemical ceramics, J. Am. Ceram. Soc. 84 (2001) 2117–2119. [7] I. Yasuda, M. Hishinuma, Mathematical analysis of stress distribution in acceptor-doped lanthanum chromites under an oxygen potential gradient, in: M. Dokiya, O. Yamamoto, H. Tagawa, S. Singhal (Eds.), Proceedings of the Fourth International Symposium on SOFC, Yokohama, 18–23 June 1995, vol. 95-1, The Electrochemical Society, 1995, pp. 924–933. [8] A. Atkinson, Chemically-induced stresses in gadolinium-doped ceria solid oxide fuel cell electrolytes, Solid State Ion. 195 (1997) 249–258. [9] H. Yakabe, M. Hishinuma, I. Yasuda, Static and transient model analysis on expansion behavior of lacro3 under oxygen potential gradient, J. Electrochem. Soc. 147 (2000) 4071–4077. [10] H. Yakabe, I. Yasuda, Model analysis of the expansion behavior of lacro3 interconnector under solid oxide fuel cell operation, J. Electrochem. Soc. 150 (2003) A35–A45. [11] R. Krishnamurthy, B. Sheldon, Stresses due to oxygen potential gradients in non-stoichiometric oxides, Acta Mater. 52 (2004) 1807–1822. [12] N. Swaminathan, J. Qu, Interactions between non-stoichiometric stresses and defect transport in a tubular electrolyte, Fuel Cells 7 (2007) 453–462. [13] A. Atkinson, T. Ramos, Chemically-induced stresses in ceramic oxygen ionconducting membranes, Solid State Ion. 129 (2000) 259–269. [14] E. Blond, N. Richet, Thermomechanical modeling of ion-conducting membrane for oxygen separation, J. Eur. Ceram. Soc. 28 (2008) 793–801. [15] A. Zolochevsky, A. Grabovskiy, L. Parkhomenko, Y. Lin, Coupling effects of oxygen surface exchange kinetics and membrane thickness on chemically induced stresses in perovskite-type membranes, Solid State Ion. 212 (2012) 55–65. [16] A. Zolochevsky, L. Parkhomenko, A. Kuhhorn, Analysis of oxygen exchangelimited transport and chemical stresses in perovskite-type hollow fibers, Mater. Chem. Phys. 135 (2012) 594–603. [17] A. Evans, J. Hutchinson, The thermomechanical integrity of thin films and multilayers, Acta Metall. Mater. 43 (1995) 2507–2530. [18] C. Hsueh, Thermal stresses in elastic multilayer systems, Thin Solid Films 418 (2002) 182–188. [19] J. Malzbender, Mechanical and thermal stresses in multilayered materials, J. Appl. Phys. 95 (2004) 1780–1782. [20] J. Malzbender, Curvature and stresses for bi-layer functional ceramic materials, J. Eur. Ceram. Soc. 30 (2010) 3407–3413. [21] C. Hsueh, E. Fuller Jr., Analytical modeling of oxide thickness effects on residual stresses in thermal barrier coatings, Scr. Mater. 42 (2000) 781–787. [22] C. Hsueh, P. Becher, E. Sun, Analyses of thermal expansion behavior of intergranular two-phase composites, J. Mater. Sci. 36 (2001) 255–261. [23] C. Chatzichristodoulou, M. Søgaard, P.V. Hendriksen, Oxygen permeation in thin, dense Ce0:9 Gd0:1 O1:95
δ membranes I. Model study, J. Electrochem. Soc. 158 (2011) F61–F72.

[24] G. Lamé, Leçons sur la théorie mathématique de l’élasticité des corps solides, Bachelier, 1852. [25] S. Timoshenko, J.N. Goodier, Theory of Elasticity, McGraw-Hill Book Company, 1951. [26] Z. Shao, W. Yang, Y. Cong, H. Dong, J. Tong, G. Xiong, Investigation of the permeation behavior and stability of a Ba0:5 Sr0:5 Co0:8 Fe0:2 O3
δ , J. Memb. Sci. 172 (2000) 177–188. [27] G.R. Gavalas, S. Liu, Oxygen selective ceramic hollow fiber membranes, J. Memb. Sci. 246 (2005) 103–108. [28] J.F. Vente, W.G. Haije, Z.S. Rak, Performance of functional perovskite membranes for oxygen production, J. Memb. Sci. 276 (2006) 178–184. [29] S. Baumann, J. Serra, M. Lobera, S. Escolastico, F. Schulze-Kuppers, W. Meulenberg, Ultrahigh oxygen permeation flux through supported Ba0:5 Sr0:5 Co0:8 Fe0:2 O3
δ membranes, J. Memb. Sci. 377 (2011) 198–205. [30] F. Schulze-Kuppers, S. Baumann, W. Meulenberg, D. Stover, H. Buchkremer, Manufacturing and performance of advanced supported Ba0:5 Sr0:5 Co0:8 Fe0:2 O3
δ (bscf) oxygen transport membranes, J. Memb. Sci. 433 (2013) 121–125. [31] Z. Yang, Y. Lin, A semi-empirical equation for oxygen nonstoichiometry of perovskite-type ceramics, Solid State Ion. 150 (2002) 245–254. [32] R. Kriegel, R. Kircheisen, J. Tøpfer, Oxygen stoichiometry and expansion behavior of Ba0:5 Sr0:5 Co0:8 Fe0:2 O3
δ , Solid State Ion. 181 (2010) 64–70. [33] R.W. Rice, Mechanical Properties of Ceramics and Composites: Grain and Particle Effects, CRC Press, 2000. [34] B. Huang, Thermo-mechanical properties of mixed ion-electron conducting membrane materials (Ph.D. thesis), RWTH Aachen University, 2010. [35] M. Lipinska-Chwalek, J. Malzbender, A. Chanda, S. Baumann, R. Steinbrech, Mechanical characterization of porous Ba0:5 Sr0:5 Co0:8 Fe0:2 O3
d , J. Eur. Ceram. Soc. 31 (2011) 2997–3002. [36] A. Kaiser, S. Foghmoes, C. Chatzichristodoulou, M. Søgaard, J. Glasscock, H. Frandsen, et al., Evaluation of thin film ceria membranes for syngas membrane reactors – preparation, characterization and testing, J. Memb. Sci. 378 (2011) 51–60. [37] M. Mogensen, N.M. Sammes, G.A. Tompsett, Physical chemical and electrochemical properties of pure and doped ceria, Solid State Ion. 129 (2000) 63–94. [38] M. Lipinska-Chwalek, G. Pecanac, J. Malzbender, Creep behaviour of membrane and substrate materials for oxygen separation units, J. Eur. Ceram. Soc. 33 (2013) 1841–1848. [39] Y. Du, N.M. Sammes, G.A. Tompsett, D. Zhang, J. Swan, M. Bowden, Extruded tubular strontium- and magnesium-doped lanthanum gallate, gadoliniumdoped ceria, and yttria-stabilized zirconia electrolytes: mechanical and thermal properties, J. Electrochem. Soc. 150 (2003) A74–A78. [40] A. Atkinson, A. Selcuk, Mechanical behaviour of ceramic oxygen ionconducting membranes, Solid State Ion. 134 (2000) 59–66. [41] Y. Touloukian, R. Kirby, R. Taylor, T. Lee, Thermal Expansion: Nonmetallic Solids, IFI/Plenum, 1977. [42] R. Tilley, Understanding Solids: The Science of Materials, John Wiley and Sons, 2005. [43] Y. Shiratori, F. Tietz, H. Buchkremer, D. Stover, Ysz-mgo composite electrolyte with adjusted thermal expansion coefficient to other SOFC components, Solid State Ion. 164 (2003) 27–33. [44] P.V. Hendriksen, J. Høgsberg, A. Kjeldsen, B. Sørensen, H. Pedersen, Failure modes of thin supported membranes, in: N. Bansal (Ed.), Advances in Solid Oxide Fuel Cells II: Ceramic Engineering and Science Proceedings, vol. 27, 2007. [45] J. Hutchinson, Z. Suo, Mixed mode cracking in layered materials, Adv. Appl. Mech. 29 (1992) 63–191. [46] S. Nemat-Nasser, H. Horii, Compression-induced nonplanar crack extension with application to splitting, exfoliation, and rockburst, J. Geophys. Res. 87 (1982) 6805–6821. [47] M. Ashby, S. Hallam, The failure of brittle solids containing small cracks under compressive stress states, Acta Metall. 34 (1986) 497–510. [48] A. Chanda, B. Huang, J. Malzbender, R. Steinbrech, Micro- and macroindentation behaviour of Ba0:5 Sr0:5 Co0:8 Fe0:2 O3
d perovskite, J. Eur. Ceram. Soc. 31 (2011) 401–408. [49] R. Mangalaraja, S. Ananthakumar, K. Uma, R.M. Jiménez, M. López, C.P. Camurri, Microhardness and fracture toughness of Ce0:9 Gd0:1 O2 for manufacturing solid oxide electrolytes, Mater. Sci. Eng. A 517 (2009) 91–96.