Mechanical Characterization of Solid Oxide Fuel Cells and Sealants

 3-11
. (
a), (b)
) or for plates via ring-on-ring, ball-on-ring or ball-on-three 
balls test (
Figure
3-11
. (
c) – (e)
) [96]. The testing of plates has become very common since 
advanced materials processing often relies on tape casting techniques [22], yielding plate type 
specimens in either dense or porous state. The latter are representative for the porous substrate 
and used in SOFCs or membrane designs [97]. For tubular geometries, which are still 
occasionally used as SOFC, testing methods such as O-ring, C-ring (
Figure
 3-11
. (
g), (h
) [98] or 
four-point bending of semi-cylindrical specimens [99] can be applied (
Figure
3-11
. (
f)
). The 
initial part of the load-displacement curve can be affected by contact problems related to 
roughness or biased by surface waviness. Hence, sometimes a grinding or polishing process is 
advantageous for plates or bars to yield materials representative curves which lead to a more 
precise determination of elastic modulus [100].
Figure 3-11: Different methods to test elastic modulus and strength in bending: (a) four-point 
bending, (b) three-point bending, (c) ring-on-ring, (d) ball-on-ring, (e) ball-on-three balls, (f) 
four-point bending of semi-cylindrical specimens, (g) O-ring, and (h) C-ring [86]. 
3) Impulse excitation test 
An advanced method to determine the elastic modulus for certain specimen geometries is the 
impulse excitation method, where the specimens’ resonance induced by a mechanical impulse is 
detected acoustically [86]. 
Figure
 3-12
 is a simple schematic of the impulse excitation technique, 
where a slight impact is induced at a side of the solid specimen.

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Literature review 
23 
Figure 3-12: A schematic of the impulse excitation technique. 
The resonance frequency is then used to calculate the elastic modulus. The value of the elastic 
modulus can be determined via [101]:
𝐸 = 0.9465 ∙ 
𝑚 ∙ 𝑓
𝑓
2
∙ 𝑡
3
𝑏 ∙ ℎ
3
∙ 𝑇 ∙ (
ℎ
𝑙
)
(3-2) 
where 
m
is the mass, 
f
f
the resonant frequency, 
l 
the length, 
b
the width, 
h
the height and 
T(h/l)
a 
geometry correction factor.
The method appears to have a higher accuracy than the four-point bending test [102]. Moreover, 
it is less time consuming when testing elastic modulus as a function of temperature. Experiments 
at elevated temperatures and under different atmospheres are possible with suitable set-ups. For 
example SOFC half-cell materials have been tested up to the typical operation temperatures 
under oxidized (NiO-YSZ) and reduced conditions (Ni-YSZ) [103]. 
In addition to the elastic modulus, this method can be also used to determine Poisson´s ratio [40] 
and the damping coefficient, latter being determined from the decay of the resonance frequency. 
This parameter gives information on the occurrence of non-elastic mechanisms (phase changes, 
viscous deformation or creep) and processes (atomic defect interactions) [104].
Problems arise in the case of materials that possess a large damping coefficient where the signal 
decays too fast and the limits of the experimental analysis can be reached. Furthermore, 
extremely thin (light) specimens can cause experimental problems due to a displacement of the 
sample during testing.

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Literature review 
24 
3.4.1.2.
Porosity influence on elastic modulus 
As well known, porosity influences the elastic modulus of materials. Many studies exist on the 
effective elastic moduli of porous solids treating them as two-phase materials, with the second 
phase being a void, and several empirical equations have been derived for describing the 
effective elastic moduli as a function of porosity [105]. The linear relation, equation (3-3) in