Introduction:

Shape memory alloys (SMAs) exhibit the extraordinary thermo-mechanical behaviors of shape memory effect (SME) and pseudoelasticity (PE) which make their constitutive modeling challenging. Considering transitions between austenite and martensite phases in these materials, Brinson (1993) proposed to separate twinned and detwinned parts of the martensite volume fraction as internal variables in 1-D models. However, in a general 3-D loading case, reorientation and phase transformation may occur simultaneously. Mehrabi et al. (2015) conducted experimental tests on thin-walled SMA samples and observed discernable axial strains under pure torsion which indicates the so-called tension-torsion coupling.

Microplane theory is an approach for 3-D constitutive modeling, based on a 1-D constitutive model, with the ability to predict reorientation under nonproportional loadings (Kadkhodaei et al. (2007), Mehrabi et al. (2012), Mehrabi et al. (2013)). To calculate integrals over a hemisphere at each integration point, Bazant et al. (1986) suggested a 21-point numerical scheme which has been vastly utilized in microplane modeling of SMAs. However, microplane models founded on this integration scheme cannot predict the tension-torsion coupling (Mehrabi et al. (2014)).

In the present work, this issue is addressed and different numerical schemes in calculating the microplane integrals are evaluated to propose an approach using which microplane formulations will be able to predict tension-torsion coupling.

Numerical Investigations

Modeling an SMA under nonproportional loadings, using microplane theory and 21-point scheme, may result in discontinues strain responses. For instance, in the tension-torsion nonproportional loading, in which first the axial stress is applied and held constant then the shear stress is applied and the stresses will diminish in the same order, a significant jump is seen at the beginning of the second step caused by the numerical integration deficiencies and change in the stress status. In the uniaxial situation, the traction on some planes is zero; but, when the shear stress is applied, these planes would have traction on them so they would affect the resulted strain tensor. At the last step, the axial strain is zero when there is only shear stress which means the inability of the microplane theory to predict tension-torsion coupling using 21-point numerical integration.

In centrosymmetric numerical schemes, such as 21-point numerical scheme, owing to the special arrangement of the microplanes, symmetric planes cancel out their values. For example, in simple shear loading, the 21-point numerical scheme would result in zero axial strain. Consequently, a non-centrosymmetric numerical scheme needs to be applied. The 25-point numeric scheme proposed by Fliege et al. (1999) showed a notable axial strain during pure shear loading which is expected.

Experimental Validation

The microplane model results, using the 25-point method, are compared with experimental findings in nonproportional tests. Comparison of the results reveals that the 25-point scheme leads to a better agreement with the experimental observations. The 25-scheme shows axial strain while the stress state is pure shear.

Conclusion

In this study, the 21-point integration scheme in microplane modeling of SMAs was studied and it was shown to yield some discontinues responses in the case of nonproportional loadings. In fact, because of the centrosymmetry of this scheme, it is not possible to reach the coupling between tension and torsion. Although the developed 25-point numeric scheme may possess sort of the same discontinuity, owing to its non-centrosymmetry, it enables the microplane formulation to predict tension-torsion coupling under nonproportional loadings.