Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved International Review of Mechanical Engineering, Vol. 5, N. 2 Special Issue on Heat Transfer
335 qualitatively the variation strain rate for which the
maximum hysteresis is reached with the geometry or the heat transfer coefficient cf. Fig. 8. Finally, the
influence of each heat source on the mechanical and thermal responses is analyzed, showing that the thermal
response is mainly governed by the latent heat term, leading to variations of the slopes of the plateaus,
whereas the mechanical dissipation is responsible for the hysteresis loop.
III.3. Cyclic Model
To the authors’ knowledge, the only work dealing with modeling both the effects of thermomechanical
coupling and training on the cyclic response of SMA is given in Morin et al. [76]. Their model is based on the
ZM cyclic model [50], coupled with the heat equation, in which the considered heat sources are latent heat and
mechanical dissipation. The mechanical dissipation has two different origins: phase change and microstructure
degradation through the creation of internal stress and residual strain. However, as in [68], the predominant heat
source is latent heat. In this model, the dissipation associated to the microstructure degradation has minor
influence on the temperature evolution. As the model assumes an homogeneous response of the wire,
temperature measurements have been averaged before being compared to the model predictions. The model is
able to quantitatively reproduce the mechanical experimental observations as well as the thermal
amplitudes and mean temperatures for strain rates ranging from 5·10
-5
s
-1
to 10
-2
s
-1
. It captures also the variation of the hysteresis area with strain rate during
cycling. The absence of the thermal expansion term in the stress-strain relation avoids from reproducing the
variation of the residual strain with strain rate. Moreover, the heat production is still assumed to be constant during
phase change, resulting in a source of discrepancy between model prediction and experimental
measurements of the temperature.
IV. Fatigue
A good understanding of fatigue in SMAs can help improve the safety of the SMA materials in industrial
applications. Two types of fatigue must be considered: classical
mechanical fatigue due to mechanical cycling in the pseudoelastic domain, and thermal fatigue or amnesia
due to a degradation of the material characteristics responsible for the shape memory effect.
Some fatigue analysis can be found in the literature [73]-[78]. In spite of their quality, the majority of these
publications is based on metallurgical considerations and remains mainly qualitative. The authors focus their
analyses on the influence of the cyclic loading on the SMA or the pseudoelastic characteristics. As a
consequence, the results are not suitable for the analysis and design of SMA structure using e.g. finite elements.
SMA fatigue criteria are mainly restricted to one- dimensional Manson-Coffin-type laws see [73], [79].
Miyazacki et al. [80] propose a splitting of the Wöhler curve into three parts, corresponding to different
dissipation regimes of the SMA: the first part, associated with the highest fatigue life is related to small strain
amplitude, and thus to an elastic behavior no dissipation; for the second part, associated with medium
fatigue lives, the phase change plateau is not attained, but a small hysteresis appears during the unloading phase.
Finally, the low cycle fatigue regime is associated with the occurrence of the martensitic transformation, which
dissipates considerable amounts of energy. Moumni et al. [81] proposed a three-dimensional criterion for SMAs.
Indeed, the mechanical behavior of SMAs under cyclic loading shows a variation of the hysteresis loop where
the hysteresis area decreases during the first cycles and then stabilizes. Taking advantage of the analogy with
low-cycle fatigue for elasto-plastic materials, a relation is established between the number of cycles to failure and
the dissipated energy at the stabilized cycle.
IV.1. Influence of the Heat Transfer on the Fatigue
Life of SMA Wagner et al. [82] investigated the influence of the
geometry of the wire and of the loading rate on the fatigue life in bending rotation fatigue experiments. They
show that the fatigue life may depend on the loading frequency, but that this dependence disappears when the
experiments are conducted in a silicon oil bath at constant temperature, or at low loading frequencies in
air. They explain these results by a self-heating of the material. Eggeler et al. [83] obtained the same
conclusions by showing that the fatigue life of SMA wires decreases when the diameter of the wire or the
loading rate increases. The dependence on these parameters disappears if the temperature of the material
is kept constant. Those two articles [82] and [83] lie in the continuity of the work presented in [84] by
Sawaguchi et al.. Tobushi et al. [62], [85] carried fatigue tests in both air and water at different loading
rates. The authors found a higher number of cycles to failure when the surrounding medium is water because of
smaller variations in temperature. A Manson-Coffin-type criterion that depends on temperature and on loading
frequency is proposed. Matsui et al. [86] perform rotating bending fatigue experiments, and show that
temperature of the surrounding, loading rate and geometry of the specimen influence the fatigue lifetime.
However, this influence depends on the surrounding medium, and is greater in the air than in the water. Predki
et al. [87] perform torsion experiments on cylindrical and tubular specimens, at different frequencies between 0.1
and 1 Hz in ambient air. No effort is made to maintain a constant material temperature and the experiments seem
to depend on different parameters, namely: the geometry
Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved International Review of Mechanical Engineering, Vol. 5, N. 2 Special Issue on Heat Transfer
336 of the specimen tubular or cylindrical specimens and
the loading path repeated or alternating torsion. The authors found no dependence on loading frequency.
IV.2. Prospects for the Study of Fatigue Life
The influence of heat transfer on SMA behavior is important for understanding the fatigue properties these
alloys and for developing a suitable fatigue criterion. Indeed, following Moumni et al. [81], the number of
cycles to failure N
f
is related to the dissipated energy W at the stabilized cycle through a power law:
f
W N
µ
δ =
4 where
δ and µ are material constants. The obtained results for tension-compression tests are shown on Fig. 9.
Fig. 9. Estimated vs. experimental lifetime of TiNi specimens subjected to tensile compressive tests at different R ratios for criterion 4. The
two black lines stand for half and twice the experimental lifetime
Following Amiable et al. [88], Morin [89] proposes to take into account the hydrostatic pressure in the criterion:
max f
W aP
N
µ
δ +
= 5
where a is a material parameter. The addition of the hydrostatic pressure improves the prediction of the
fatigue life, as shown on Fig. 10.
Fig. 10. Estimated vs. experimental lifetime of the same specimens as Fig. 9 for criterion 5. The two black lines stand for half and twice the
experimental lifetime
Most of the tests presented in the different articles are strain controlled. It follows that the higher the
temperature due to self heating at high frequency for example, the higher the maximum stress. Thus, a
criterion based on dissipated energy and maximum hydrostatic stress coupled with a thermomechanically
coupled model could account for the frequency effect. Since the evolution of the dissipated energy is non
monotonic with respect to the strain rate, a non monotonic evolution of the fatigue lifetime can be
expected, with a minimum corresponding to maximum hysteresis area, since the fatigue of SMA structures
depends on the loading frequency and on the wire diameter. This work is undertaken and will be presented
in a future publication.
V. Conclusion
