## 5 Discussion of Results  
A cursory inspection of the data in Figures 3 and 4 indicates the longitudinal extrusion process results in (a) circumferential symmetry in the material response for both MB1 and MB2 specimens, (b) a greater than 20% increase in tensile yield stress for the MB2 specimen, and (c) a reduction in shear yield stress by 12% for the MB2 specimens. The measured increase in tensile yield stress is nominally consistent with manufacturer data shown in Table 1. Interestingly, the radial tensile results for both MB1 and MB2 specimens are quite similar. The effect of anisotropy in the behavior of both longitudinally extruded Al6061-T6 cylindrical tubes is evident in the values of the six Yld61 parameters in Table 3 for both MB1 and MB2 bars. Several of the parameters show significant deviation of the six anisotropic parameters from unity, where unity is the value required for isotropic yielding. Though there are slight changes in elemental composition shown in Table 1, the most likely source of the different behavior for MB1 and MB2 specimens is the longitudinal extrusion process that resulted in significant changes in material microstructure that are shown in Figure 1. As noted previously, the large difference in microstructure for MB1 and MB2 mother bars (Figure 1) shows that the MB2 specimen has undergone much higher elongation and transverse compression than MB1 during the \"nominally similar\" extrusion processes. Such differences are consistent with the observed increase in work-hardening that are shown in Table 4, as well as the higher yield stress and higher ultimate stresses shown in Table 1 for MB2 relative to MB1. As shown in Figure 5, the Barlat Yld91 [17] predictions for tensile loading in the radial direction (X) and axial direction (Y) are in excellent agreement with the experimental data, whereas the Von Mises yield criterion overpredicts the required yield stress for loading in the radial direction by up to 25%. The source of the relatively strong yield anisotropy is most evident in the Figures 3 and 4, where yield stresses in radial directions for the MB2 material is \(\sim\)25% lower than in the longitudinal direction (270 Mpa vs. 340 Mpa). Furthermore, as shown in Figure 5, the effect of yield anisotropy is largest for \(\sigma_{\mathrm{x}}/\sigma_{\mathrm{y}}\approx 0.5\), with the Barlat criteria predicting yielding for lower stresses (15% for MB1, 30% for MB2) than the Von Mises criteria. Thus, the Barlat yield criteria provides substantially improved accuracy in the prediction of yielding for those applications where such differences are truly important. As shown in Figures 8 and 9, prediction of the torsional specimen response during loading shows that (a) isotropic yield models underestimate by \(\sim\)10% the experimental  
Figure 9: Comparison of experimental and theoretical shear stress vs. total shear strain response for Al6061-T6, MB2. results for MB1 and substantially over-estimate the response by \(\sim\)25% for MB2. When using the Barlat YId91 anisotropic yield criterion, the deviation is less than 5% for both specimens, again demonstrating the importance of anisotropy for accurate prediction of material response in extruded material systems. This observation is consistent with previous studies by [PERSON] et al. [27] and [PERSON] [28]. Regarding the choice of yield criteria, the investigators selected the Barlat YId9 [17] anisotropic yield function with six material parameters (rather than the YId2000-2d, YId 2004-8d or YId2004-18p [20,21] models) for two reasons. First, the diameter of two mother bars was 28.575 mm, which was relatively small, limiting the ability to extract usable specimens from different directions for parameter identification. Secondly, the longitudinal extrusion process of the Al6061-T6 tubes develops similar material response in all circumferential directions, limiting the number of required specimen orientations necessary for anisotropic model calibration.