## 4 Theoretical Prediction and Experiments for Simple Torsion Case  
Once the elastic, hardening, and anisotropic yielding parameters are obtained for the two extruded bar materials, the theoretical formulae given in Section 2 can be com  
\begin{table}
\begin{tabular}{c c c} \hline
**Material Properties** & **6061-T6 Al Tube (MB1)** & **6061-T6 Al Tube (MB2)** \\ \hline Modulus of Elasticity & 69 Gpa & 69 Gpa \\ \hline Poisson’s ratio & 0.33 & 0.33 \\ \hline Hardening Parameter, n & 13.51 & 16.47 \\ \hline \end{tabular}
\end{table}
Table 4: Elastic properties and power law hardening parameters.  
Figure 5: Predicted normalized yield surfaces for biaxial stress states with Barlat and Von Mises yield function for two Al 6061-T6 bar.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline
**6061-T6 Al Tube** & **m** & **a** & **b** & **c** & **f** & **g** & **h** \\ \hline MB1 & 8 & 1.0000 & 1.1516 & 1.000 & 0.8750 & 1.0835 & 0.8750 \\ \hline MB2 & 8 & 1.0000 & 1.4452 & 1.000 & 1.2069 & 1.3059 & 1.2069 \\ \hline \end{tabular}
\end{table}
Table 3: Anisotropic yield function parameters for longitudinally extruded Al6061-T6 rods.  
bined to predict the complete history of elastic-plastic deformations in thin-walled tubular specimens that are subjected to simple torsional loading.  
Figure 6 presents a flow chart for the numerical program that was developed and implemented by the authors to predict the elastic and plastic strains. Every step, the program is determined effective stress according to [PERSON]'s and von [PERSON]'s yield function presented in Equations (5) and (6) respectively and then compare it with reference yield stress state. If the effective stress is below the yield stress, the model only determines elastic shear strain (Equation (3)) which is equal to total shear strain. After yielding, incremental plasticity is combined with the elastic equations to predict the total shear strain (Equations (2)-(4) and (8)) for specimens subjected to known applied shear stresses. The input load cell data is presented in Figure 7. The accuracy of the predictions is assessed by direct comparison to the experimental measurements for the total shear strain on the specimen surface during torsional loading using the same loading path as shown in Figure 7. For that, two simple torsion experiments were conducted using MB1 and MB2 tubular specimens (Figure 10).  
Figure 6: Flow chart theoretical prediction of total shear strain using isotropic and anisotropic yield criteria.  
As discussed in Appendix A, all surface strain measurements were obtained using a stereovision system with VIC-3D software employed to analyze the images. Baseline studies using images at zero load showed the standard deviation in the measured surface effective strain is \(\leq\)100 \(\mu\epsilon\). During torsional loading, variability in the measured surface strains increased with the signal-to-noise ratio \(\geq\) 16 for all torsional loads, confirming that the measured surface strains are within \(+/-6\%\) of the average value used for comparison to the theoretical predictions.  
Figures 8 and 9 present direct comparisons of the experimental shear stress vs. total shear strain measurements to theoretical predictions using isotropic and anisotropic yield criteria for MB1 and MB2 specimens, respectively. Inspection of the results shows that the model using the Barlat anisotropic yield function is in very good agreement with experimental measurements, with differences less than 5% for both specimens. Conversely, predictions using a Von Mises isotropic yield function show deviations from the experimental data ranging up to 10% and 25% for the MB1 and MB2 specimens, respectively.