## 6 Summary and Conclusions  
In this study, the elastic, plastic, and total shear strain components are obtained (a) theoretically through computational modeling using both the Von Mises and Barlat YId91 yield criteria with incremental plasticity, and (b) experimentally using StereoDIC for extruded, thin-walled tubular specimens subjected to applied torsional loads. Results from the experimental studies and analytic modeling are summarized as follows:  
* Uniaxial tension and simple torsion experiments were performed on a series of specially machined tubular and dog-boned rectangular specimens extracted from two longitudinally extruded Al6061-T6 cylindrical bars to obtain the required material parameters for anisotropic yield criteria. Here, the yield function YId91 developed by [PERSON] et al., with six measured material parameters, is used to model the anisotropic response of the extruded aluminum material and an isotropic yield criterion using the Von Mises criteria is employed to predict an isotropic response. * Since the extrusion process is circumferentially symmetric for our longitudinal extrusions, yield stresses in all radial directions are expected to be similar, a condition that was confirmed from a series of radially oriented specimen experiments. * Assuming isotropic strain hardening beyond yielding, the Von Mises isotropic and Barlat YId91 parameters yield functions with power-law hardening and incremental theory of plasticity are used to develop and then implement a constitutive model for elastic-plastic material behavior and predict the response of the extruded MB1 and MB2 material. These constitutive models are implemented in a relatively simple numerical analysis platform to predict the stress-strain response for both materials undergoing torsional loading. * Theoretical results indicate that results obtained using the anisotropic yield function are in excellent agreement with simple torsion experiments for both materials. Conversely, results obtained using an isotropic yield function underestimate, and then over-estimate, the stress-strain response of the MB1 and MB2 specimens, respectively. * Direct comparison of the experimental and theoretical results indicates that both extruded Al6061-T6 materials are significantly anisotropic, with the longitudinal yield stress deviating from the radial yield behavior in the extruded materials by over 20%. Furthermore, for biaxial loading cases, the Barlat yield criteria provides improved accuracy in the prediction of yielding for those applications where isotropic yielding is not adequate. * Predictions of the stress-strain response for the MB1 and MB2 tubular specimens using the Barlat YId91 anisotropic yield criterion are in excellent agreement with experimental measurements, with differences less than 5% for both specimens, clearly demonstrating the importance of yielding anisotropy for accurate prediction of material response when undergoing complex manufacturing processes, including longitudinal extrusion. * Results also show that the nominally similar extrusion processes for mother bars MB1 and MB2 are not the same, with substantial differences in the stress-strain behavior for the two materials being consistent with microstructural features that indicate the MB2 extrusion process was more severe. In this article, the formal analysis was done by [PERSON] and [PERSON]. The detailed methodology was provided by [PERSON], [PERSON], [PERSON] and [PERSON]. The theoretical and numerical analysis was done by [PERSON] The experiment was conducted by [PERSON] and [PERSON], [PERSON] prepared the original draft; review and editing were done by [PERSON], [PERSON] and [PERSON]. All authors have read and agreed to the published version of the manuscript. The authors would like to acknowledge the financial support provided by the Department of Mechanical Engineering at the University of South Carolina.