1.2 Literature Review
1.2.2 Modeling of SMA Behavior
are redefined using a new internal state variable. The martensite phase is divided into two components, as temperature induced or twinned martensite (ξT), and stress induced or de- twinned martensite (ξS). Thus, the total martensite fractionξ can be given asξ = ξS +ξT. Khandelwal and Buravalla [30] modifies the phase kinetics, so as to overcome the problem of Brinsons model. In that model under certain conditions i.e. in the dual transformation zone where the austenite phase is transformed to both temperature and stress-induced martensite, the martensite volume fraction exceeds 1; which is not possible, as ‘ξ’ can vary only from (1 to 0) for complete reverse transformation and (0 to 1) for forward transformation. This hap- pens because of the independent kinetics of twinned and detwinned martensite as proposed by Brinson [29]. The two main assumptions of the model are
• There should be a coupling between the evolution of twinned and detwinned fractions.
• The twinned martensite fraction can in turn convert into the detwinned fraction due to loading, whereas the detwinned fraction cannot convert to twinned fraction through a decrease in temperature (irreversibility of detwinning transformation).
Buravalla and Khandelwal [31] further modified the Brinson model [29] to address the discrepancy between the differential and integrated form of the constitutive equation of SMA. In this model, the transformation tensor (Ω) has been modified as,
Ω(ξ, ε) =−lD(ξ) + (−lξS)(DM −DA). (1.1)
Accordingly, the differential form of the constitutive equation has been modified and fol- lowing which the original integrated form can be derived. The above defined models are applicable for complete transformation, where the load are either monotonically increases or decreases, so as to complete the transformation. But there are cases where, within the trans- formation zone the phase transformation may not be continuous during thermo-mechanical
(stress and temperature vary simultaneously) loading. In the case of partial transformation, where the load may vary arbitrarily, different models are proposed, to capture the SMA be- havior during arbitrarily loading and unloading.
It has been noted that the martensite volume fraction (ξ) does not depend only on the current stress and temperature, but it also depends on the loading history. To simulate this Bekker and Brinson [32] propose a mathematical model for any thermo-mechanical loading.
According to this the phase transformation at a point of loading path occurs only if the direc- tion of loading path has a component in the direction of transformation. It has been proposed that the amount of phase transformation is proportional to the distance traveled along the loading direction. A switching point has been introduced, distinguishing the portion of the path, where the transformation is active, from the rest of the loading path where transforma- tion is inactive. Buravalla and Khandelwal [33] also proposed a path dependency (memory) approach for modeling the behavior of SMA during arbitrarily thermo mechanical loading.
According to this the memory within the transformation zone, is defined as the minimum distance on the loading path from the finish transformation boundary of all the points pre- viously traversed. Moreover, within the transformation zone, the transformation will occur only if the distance between the point of interest and the finish boundary is less than the memory parameter, otherwise the transformation will not happen.
Banerjeeet al[3], proposes a algorithm for updating the above defined memory param- eter, when dealing with time varying voltage. According to the above discussed model [32], the transformation in a particular zone is active only if the loading path has a component in the direction of transformation. Hence for this case, the priori knowledge of the loading path is important. But for the cases, where the loading path is not known beforehand, e.g., in a SMA actuated beam or spring biased system. Here, a time varying voltage is applied across the SMA wire to deform and undeform the beam or spring. It has been shown that in these
cases the state of transformation depends on the rate of the temperature in a given transfor- mation zone. The authors also discusses the algorithm to determine the transformation zone as well as to update the memory parameters. These are very useful so as to implement the Bekker and Brinson [32] model.
All the above defined constitutive equation was derived from the principles of thermo- dynamics taking martensite fraction as an internal variable. Whereas, Li.et al[34] proposed a constitutive model in strain form during both temperature and stress loading process as
ε =εR+εE+εT +εP
=εresξSM + σ
(ξSMESM+ξT MET M+ξAEA)
+ (ξSMηSM +ξT MηT M +ξAηA) (T −T0) +εP
(1.2)
Here, the first term in the RHS is the recoverable strain, the second term denotes elastic strain and third and fourth term represents the thermal expansion and plastic strain respectively.
ESM, ET M and EA denote the elastic moduli of detwinned martensite, twinned martensite and austenite, respectively. ηSM,ηT M andηArepresents the coefficients of thermal expansion in the respective phase. Here the martensite volume fraction of twinned martensite (ξT M), de- twinned martensite (ξSM) and austenite (ξA) phase is determined experimentally using DSC and uniaxial tension test. For this three crystal structures, six different phase transformation processes are proposed.
Researchers have also modified the above mentioned models so as to avoid the dis- crepancies or complexities depending on the applications [35, 36]. Mathematical model of shape memory alloy that can take care of the hysteretic behavior of SMA is developed in [37]. The thermodynamic model based on free energy and dissipation potential is presented in [38]. The main shortcoming of this model is that it requires a large number of material parameters, which in turn increases the computational time and hence, restricting the model
for real time applications.