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Table
3-2
, was first used by Fryxell and Chandler [106]. The exponential relation, equation (3-4),
has been used frequently and was initially proposed by Spriggs [107]. Hasselman [108]
suggested a non-linear relation , equation (3-5), which considered the voids as a dispersed phase.
Some theoretical expressions have been obtained from more fundamental analytical models,
which encompass the effect of shape, volume, distribution and interaction of pores on the elastic
constants [109-111]. However, analytical procedures considering the two-phases as continuum
materials are not directly applicable in cases where the second phase is a void. Ramakrishnan
and Arunachalam [112, 113], equation (3-6), developed a theoretical approach for determining
the effective moduli of porous solids with randomly distributed pores on the basis of the
composite sphere method (CSM). Phani and Niyogi [114] proposed on the same basis equation
(3-7).
Table 3-2: The relations between porosity and elastic modulus.
Equations
0
(1
)
E
E
mP
(3-3)
0
exp
E
E
bP
(3-4)
0
(1
)
1 (
1)
AP
E
E
A
P
(3-5)
2
0
(1
)
1
E
P
E
E
k P
(3-6)
0
(1
)
n
E
E
aP
(3-7)
The pores not only affect the elastic behavior of the materials, but also influence other
mechanical properties such as fracture strength, fracture toughness, creep, etc.
Literature review
25
3.4.2. Fracture strength
Due to the importance of giving a loading/stress limit for application, fracture strength is one of
the most commonly cited properties for ceramic materials. In the SOFC system, a high
mechanical reliability is needed [115]. Strength characterization is necessary to validate and
optimize the quality of the production of cells [116]. A number of techniques and methodologies
have been developed for the measurement of fracture strengths. Compressive strength of
ceramics is generally higher than their tensile strengths; most of these techniques equate the
fracture strength to the maximum tensile stress at fracture.
For bar-shaped specimens, three-and four-point bending tests (
Figure
3-11
(
a, b)
) are the
effective methods to determine the fracture strength, however, a careful preparation of the edges
and surfaces is required to gain the material’s representative properties [117]. Advanced set-ups
can permit the measurement of entire specimen series even at elevated temperatures [116].
During the tests, some modified specimen geometries, such as head-to-head specimen of sealant,
are widely used to mimic the real operation environment in SOFC stack, which provide more
information [118]. Ring-on-ring, ball-on-ring, ball-on-3-balls or pressure-on-ring set-ups (
Figure
3-11
(
c – h)
) are widely used on plate-shaped specimens. Due to the thin plate-sharped SOFC
cell layers, these testing methods show clear advantages. Here the highest stress occurs within
the central part of the specimen and the edges are under very low and negligible stress, which
release the requirement of the edge preparation. Layered composite plates can show curvature
effects. In the case of wavy specimens, the ball-on-three ball method was shown to be
advantageous [117].
To mimic the real case in the SOFC stack, a joined specimen so called head-to-head specimen
has been successfully used to determine the fracture stress in Jülich [82, 118].
Figure
3-13
shows
a schematic of the bending test on head-to-head specimen. A complementary statistical Weibull
analysis yields the characteristic fracture strength and the Weibull modulus, which gives
information on the fracture stress distribution. Characteristic strength and Weibull modulus can
be used to determine and predict the failure probability for a given stress state [119].
Literature review
26
Figure 3-13: Head-to-head specimen loaded in a 4-point bending test.
The sealant material in an SOFC stack is exposed to tensile and shear stresses [120]. Hence, the
characterization of shear strength is an issue and several mechanical tests have been developed
and used, but few standard tests are available and universally accepted. The asymmetrical four
point flexural test (ASTM C 1469-10, as shown in
Figure
3-14
(
a
)) [121] is recommended to
measure shear strength of ceramic butt joints, however, there are some serious shortcomings of
these testing methods, such as difficult sample preparation or a minimal displacement and
difficult avoidance of misalignments in the test set-up. Commonly adopted test methods like
single lap shear test (
Figure
3-14
(
b)
) do not measure shear strength properly: they give
“apparent shear” resulting from a mixed state of stress included shear, bending, and tensile
stresses [122]. FE simulation indicates that the stress distribution in the middle of the joint also
combines a shear stress and a traction normal stress [122]. A lap based test (ISO13124 standard)
with a cross-bonded specimen faces a similar issue, which can therefore only provide an
apparent shear strength, as shown in
Figure
3-14
. (
c)
[123].
Literature review
27
(a)
(b)
(c)
Figure 3-14: Shear testing methods (dimension mm): (a) asymmetrical four-point bending test;
(b) single-lap test in compression (SL)(c) cross-bonded test
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