Results

For preliminary verification of the model described in the preceding section, the model has been implemented into a MATLAB program for rapid prediction of the effects of cyclic loading on the behavior of an SMA. Utilizing the modified constitutive model developed in the previous section, it is now shown how this model captures the damage behavior of the SMA. The material parameters used for the simulations are shown in Table 5.2, in addition to the damage parameters previously shown in Table 5.1.

Based on these parameters, it is possible to predict the evolution of damage within a SMA subjected to cyclic thermal actuation. For a SMA actuator subjected to a constant 400 MPa load,

Table 5.2: Material Parameters Parameter Value EA 80 GPa EM 60 GPa AS 200◦C AF 215◦C MS 175◦C MF 155◦C CA 7 M P a◦_C CM 7 M P a◦_C αA 2.2x10−5 αM 2.2x10−5 Hmin 0 Hsat 0.028 k 0.0172 MPa−1 ¯ σcrit 120 MPa

the damage accumulation predicted is shown in Fig. 5.4. Due to the direct impact of damage on the elastic portion of the total Gibbs free energy, the evolution in damage in turn has a direct impact on the effective modulus as well as the elastic strain as shown in Figs. 5.5 and 5.6, respectively.

In addition to demonstrating a good fit between the evolution of damage as determined from the numerical and experimental results, it is also useful to demonstrate that the proposed model is capable of predicting the actuation fatigue lifetime of SMA actuators under multiple loading con- ditions. Such a comparison is provided in Table 5.3. As can be seen, the proposed implementation is capable of predicting the actuation fatigue lifetime not only for constant load conditions, but is also capable of predicting the actuation fatigue lifetime for SMA actuators subjected to variable loading conditions. Furthermore, the model is shown to be able to predict the actuation fatigue lifetime with a better match to experimental results in most cases as compared to previous work which utilized the fatigue life indication parameter method as discussed by Wheeler [87].

After initial confirmation of the suitability of the proposed model to capture the evolution of internal damage, the full model has also been implemented into a user material subroutine (UMAT) for use in the finite element modeling software ABAQUS. Utilizing the UMAT developed based

Figure 5.4: Evolution of damage during actuation fatigue lifetime in a SMA actuator subjected to 400 MPa tensile load.

Figure 5.5: Evolution of effective modulus during actuation fatigue lifetime in a SMA actuator subjected to 400 MPa tensile load.

Figure 5.6: Evolution of elastic strain during actuation fatigue lifetime in a SMA actuator subjected to 400 MPa tensile load.

Table 5.3: Comparison of predicting and experimental actuation fatigue lifetimes for multiple loading conditions

Load Path Min Stress Max Stress Experimental Cycles Predicted Cycles Prior Work

(MPa) (MPa) to Failure to Failure

Constant 200 200 21258 19116 23432

Constant 300 300 9742 7826 8126

Constant 400 400 4889 4869 4581

Linear 300 400 6605 6521 6357

Figure 5.7: Uniaxial truss element actuation fatigue modeling test specimen

on the Lagoudas et al. model from 2012 [95], the necessary modifications to this UMAT were completed in order to include damage based on the equations derived in Sec. 5.1.2. In terms of implementation, the UMAT assumes that the local damage from the previous time increment applies to the current time increment, and the local damage is updated as an additional subfunction at the end of the UMAT. For verification that the implementation of the damage model in the UMAT was correctly completed, two test cases were run. These test cases were for a simple 1 element uniaxial truss as shown in Fig. 5.7.

The uniaxial truss element was simulated first in order to verify that the UMAT would run prop- erly. As mentioned, the damage model was implemented into the UMAT through modification of an existing UMAT based on the SMA constitutive model from Lagoudas et al. [95]. Therefore, after the necessary modifications for the damage model were introduced, this simple uniaxial truss model verified that the model could still run properly, and that the quantities of interest for the damage model were properly captured. Specifically, in order to utilize the implementation in order to determine actuation fatigue life for arbitrary shapes and loading paths, it was necessary to deter- mine the evolution of damage throughout the entire specimen. For the uniaxial truss element, this means that the damage at all points should evolve in the same manner. Therefore, in order to allow for modeling of the uniaxial truss element under conditions of interest, and in order to be compara-

Figure 5.8: Uniaxial truss element damage after 1200 thermal ac- tuation cycles subject to 400 MPa

Figure 5.9: Evolution of damage in uniaxial truss element specimen over entire actuation fatigue lifetime subject to 400 MPa

ble with the experimental results gathered in Ch. 4 as well as in other actuation fatigue works, the top side of the specimen was fixed, while a pressure load was applied to the bottom face, resulting in a stress of 400 MPa throughout the specimens. After application of the load, the temperature was cycled from 300◦C to 150◦C, as such mimicking the experimental actuation fatigue cycling conducted at 400 MPa. Indeed this was captured correctly in that the damage throughout the entire specimen evolved in the exact same way, as shown in Fig. 5.8 after 1200 cycles. Similarly, it was necessary to ensure that damage evolved in the expected non-linear manner, which is captured for the entire actuation fatigue lifetime in Fig. 5.9.