5.3 Modelling the stiffness degradation by using Thermoelastic
Data: A calibration by using mechanical data
Fig. 9a reports the(S1_2prc)N /(S1_2prc)N0 data plotted versus N while in Fig. 9b, the data are compared
in terms of N/Nf . In Fig. 9a, the thermoelastic
data show a stress dependence from lower to higher imposed stress. Such
stress effect is slight when data are compared in terms ofN/Nf , Fig. 9b.
By observing both Fig. 9a-b, it appears that mechanical and thermal data
follow the same trend. In particular, the thermoelastic signal metrics
exhibits a steady state from N/Nf =0.4 at a value
of 0.4. (Unit Signal/ Unit Signal).
In Fig. 10, the data of E/E0 and(S1_2prc)N /(S1_2prc)N0 are reported in the same plot for each test as function ofN/Nf . The extensometer data in terms ofE/E0 have been opportunely sampled, in order to
compare mechanical and thermal data at the sameN/Nf values. In Fig.10a-b-c-d, a slightly higher
sensitivity of thermoelastic data than mechanical data is observed: for
each stress level the initial stiffness loss described by(S1_2prc)N /(S1_2prc)N0is characterised by a more severe decrease. This can be attributed to
the fact that the thermoelastic signal provides a local information of
the behaviour if compared to the mean value of the stiffness provided by
extensometer.
The following step was the correlation of thermal and mechanical data at
fixed N/Nf values. By using for(S1_2prc)N /(S1_2prc)N0 the same mathematical model used for E/E0 , we
obtain Eq. (9):
\(\frac{({S1\_2\text{perc})}_{N}}{{(S1\_2\text{perc})}_{N0}}=\left(C^{\prime}\sigma_{\max}\right)^{a^{\prime}}\left(\frac{N}{N_{f}}\right)^{b^{\prime}}\)(9)
where the symbols a’ , b’ and C’ are empirical
constants.
By including Eq. (9) in Eq.(8), a direct relation betweenE/E0 and(S1_2prc)N /(S1_2prc)N0 can be obtained, Eq.(10).
\(\left(\frac{E}{E_{0}}\right)_{\text{mod}}=\left(C\sigma_{\max}\right)^{a}{\left(\frac{1}{C^{\prime}\sigma_{\max}}\right)^{a^{{}^{\prime}}k/b^{\prime}}\left(\frac{({S1\_2\text{perc})}_{N}}{{(S1\_2\text{perc})}_{N0}}\right)}^{k/b^{\prime}}\)=\(\overset{\overline{}}{A\ }{\ \left(\frac{({S1\_2\text{perc})}_{N}}{{(S1\_2\text{perc})}_{N0}}\right)}^{\overset{\overline{}}{b}}\)(10)
Eq. (10) represents a mathematical relation between stiffness
degradation and thermoelastic signal. It shows also that to obtainE/E0 values from thermoelastic data one can:
- To find the coefficients \(\overset{\overline{}}{A}\) and\(\overset{\overline{}}{b}\) indirectly estimating a , k ,a’ , b’ , C , C’ .
- To find \(\overset{\overline{}}{A}^{\prime}\) and
‘\(\overset{\overline{}}{b}\)’ directly by plottingE/E0 versus(S1_2prc)N /(S1_2prc)N0 .
The coefficients are reported in Table III for each stress level. They
are slightly similar at each stress level, as confirmed by the
experimental data (both mechanical and thermal), Fig.11.
In Fig. 11, for each stress level the E/E0 data
are reported compared to(S1_2prc)N /(S1_2prc)N0 data. Specifically, in Fig.11a-b, the high number of acquired data
close to the value 0.80 of E/E0 , is due to the
presence of a plateau of stiffness degradation. The test run at the
stress level of 70%UTS, Fig. 11d, is the one with a smaller number of
thermal acquisitions, but they are well distributed for each value ofE/E0 .
An advantage of the present material is such that it does not present a
marked stress dependence when the metrics E/E0and (S1_2prc)N /(S1_2prc)N0 are compared to N/Nf . So, in order to describe
the material behaviour one can choose the coefficients\({}^{\prime}\overset{\overline{}}{A}^{\prime}\) and ‘\(\overset{\overline{}}{b}^{\prime}\) of a
specific stress level from Table III. If the test at 70%UTS is adopted
as representative of the relation between thermal and mechanical data,
the model is:
\(\left(\frac{E}{E_{0}}\right)_{\text{mod}}\)=\(1.01{\ \left(\frac{({S1\_2\text{perc})}_{N}}{{(S1\_2\text{perc})}_{N0}}\right)}^{0.22}\)(11)
In the next section, the validation of the model for the data at
different stress levels will be performed and an estimation of errors
between measured data and modelled data is provided.
Discussion
In this section (E/E0 )modvalues obtained by Eq. (11) have been expressed as a function ofN/Nf by using a power law in of Eq. (4) where the
coefficient depending from stress \(K\left(\sigma_{\max}\right)\ \) is
0.81 and the exponent k is -0.03.
In Fig.12 are reported for each stress level, the experimental data
compared with the models according to Eq. (3)-(8) and the model obtained
by calibrating thermoelastic data(E/E0 )mod .
The results in Fig. 12 show a promising correlation between the
stiffness degradation obtained from thermoelastic data-based model and
extensometer data. In particular, at stress levels higher than the
endurance limit (Fig. 12b-c-d) the(E/E0 )mod data match the experimental data at the steady state better than the
other models.
The capability of the proposed approach can be also assessed by
evaluating absolute errors between the values forecast by each model and
experimental data (E/E0 ), at each stress level in
terms of maximum stress:
\(\text{Absolute\ Error}_{1}=\left|{\left(\frac{E}{E_{0}}\right)_{Eq.3}-\left(\frac{E}{E_{0}}\right)}_{\frac{N}{N_{f}}}\ \right|_{\sigma_{\max}}\)(12)
\(\text{Absolute\ Error}_{2}=\left|{\left(\frac{E}{E_{0}}\right)_{Eq.8}-\left(\frac{E}{E_{0}}\right)}_{\frac{N}{N_{f}}}\ \right|_{\sigma_{\max}}\)(13)
\(\text{Absolute\ Error}_{3}=\left|{\left(\frac{E}{E_{0}}\right)_{\text{mod}}-\left(\frac{E}{E_{0}}\right)}_{\frac{N}{N_{f}}}\ \right|_{\sigma_{\max}}\)(14)
where (E/E0 )Eq.3 ,(E/E0 )Eq.8,
(E/E0 )mod are respectively the
stiffness degradation modelled using Eq.(3), Eq.(8) for just mechanical
data and Eq. (11) for thermoelastic data.
The results reported in Fig.13b demonstrate as the model described by
Eq.(8) reproduces in a better way the experimental behaviour at specific
stress level if compared to the model of Eq. (3), Fig.13a.
In the case of Fig.13b, the absolute error reduces asN/Nf increases while is it is high at initial
cycles.
Fig.13c represents the absolute error evaluated between the values of(E/E0 )mod and experimental
data. The Absolute Error3 is comparable with the
other two estimated errors specifically, it is generally lower thanAbsolute Error1 . This demonstrates the capability
of thermoelastic data in describing the stiffness degradation of the
material.