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(3-17)
(3-18) and (3-15), respectively. To analyze the porosity effect
numerically, the creep rates obtained experimentally are compared with the analytical models in
Figure
5-20
. The analytical models were used to model the creep rates at 800 ºC under 10 MPa,
in fact, the tendency of the models was similar for the other data sets. Among the analytical
models, the Hashin-Shtrikman yielded the good approximation in range of 50-70% equivalent
porosity, while the Ramakrishnan model provides the closet prediction when the equivalent
porosity is above 70% in agreement with [44].
Figure 5-20: Comparison of the creep rates obtained from analytical models and ring-on-ring
bending tests at 800°C.
5.1.3.4.3.
Comparison of 4-point and ring-on-ring bending creep
for material C
Once the stress exponent is determined (see section 5.1.3.3), the creep rate of 4-point bending
test can also be determined by equation (3-12). A comparison of creep rates of 4-point and ring-
on-ring creep along with reference data is shown in
Figure
5-21
.
Results and discussion
101
The literature results [140] are slightly higher than the values obtained in this work, the reason
might be assigned to material difference and/or experimental uncertainty. Creep rates of material
C obtained from ring-on-ring tests are around 3 times lower than the values obtained from 4-
point bending tests. This effect might be assigned to 1) stress distribution changes in 4-point
bending and ring-on-ring bending tests and also 2) the limitation of analytical equation used for
ring-on-ring creep. As mentioned above, the equations used for ring-on-ring creep calculation
assumes a stress exponent of 1, which is not the case for the material in the current study. Since
this could be the origin of the differences between 4-point bending and ring-on-ring creep, the
accuracy of ring-on-ring creep determination is discussed in the following subsection.
Figure 5-21: Comparison of creep rates obtained via 4-point and ring-on-ring tests along with
reference data [140].
5.1.3.5.
FEM simulation
A FEM analysis was carried out to analyze the origin of different creep behavior in ring-on-ring
and 4-point bending tests, creep parameters derived from 4-point bending test in this work were
used as input for the simulation (
Table
4-9
in section 4.3.3). In the FEM simulation the creep
rates were derived in two ways, i) from the equivalent creep strain by ANSYS and ii) from the
Results and discussion
102
deflection using the simulated displacement with the analytical formulas (Equation (3-12) and
(4-5)). The creep rates obtained by these two methods are termed “FEM Result” and “FEM
Equation”, respectively, in the following. The creep strain at the bottom surface of the specimen,
which is rather constant within the area enclosed by the loading ring, was taken as the FEM
numerical result. An example of the simulated creep strain under 30 MPa at 800ºC by 4-point
bending test is shown in
Figure
5-22
.
Figure 5-22: The equivalent creep strain simulated by ANSYS for a 4-point bending test under
30 MPa at 800 ºC.
The comparison of simulations and experimental data is shown in
Figure
5-23
. Both simulation
results for 4-point bending creep (FEM Result and FEM Equation) show an agreement with
experimental results at rather low stresses (10 and 15 MPa). The experimental value is slightly
lower (the difference is smaller than 9%) than the simulation values which might be caused by
friction during the real testing. The friction is not taken into account in case of simulation, while
it seems that friction has an effect on creep rate. However, the FEM Result and FEM Equation
also agree with each other (the difference is smaller than 4%), indicating the rather accurate
analysis for creep in 4-point bending tests.
The FEM Equation and experimental data of ring-on-ring bending creep, both based on
formulation, show general agreement. Similarly the friction effect might play an important role
for this difference, which could have a stronger effect on ring-on-ring test due to the larger
contacted area. While the FEM Results yielded around 50% and 72% higher creep rate compared
to values of FEM Equation and experimental result, respectively, indicating the limitation of an
Results and discussion
103
analytical formulation to analyze data from ring-on-ring tests (Equation (4-5)). The formula
analysis neglected changes of geometry, stress distribution due to the creep deformation and
assume the unit stress exponent. It seems that the neglected changes of stress distribution and
geometry play an important role in an accurate determination of ring-on-ring test creep rates.
To assess the inaccuracy caused by stress exponent in ring-on-ring equation
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