## 2 Background Theory for Yield Functions and Plastic Strain  
Since the initiation of macroscopic plastic deformation is associated with the concept of material yielding, the ability to predict yield for a range of multi-axial stress states is an essential component of the analysis and design processes. A general form for the yield function can be written:  
\[F\big{(}\sigma_{ij},\bar{\epsilon}^{p}\big{)}=\overline{\sigma}\big{(}\sigma_ {ij}\big{)}-\kappa(\bar{\epsilon}^{p}) \tag{1}\]  
where \(\overline{\sigma}\big{(}\sigma_{ij}\big{)}\) represents the effective stress for a given stress state, \(\sigma_{ij}\), \(\kappa(\bar{\epsilon}^{p})\) is the hardening rule derived from uniaxial tension data and \(\bar{\epsilon}^{p}\) is the equivalent plastic strain. Employing incremental plasticity, an increment of total strain (\(de_{ij}\)) for any strain component at an arbitrary point can be expressed as the sum of an elastic strain increment (\(de_{ij}^{e}\)) and a plastic strain increment (\(de_{ij}^{p}\)) as follows:  
\[d\epsilon_{ij}=d\epsilon_{ij}^{e}+d\epsilon_{ij}^{p} \tag{2}\]  
The elastic strain increments in Equation (2) can be determined from [PERSON]'s law, with the elastic response of the tubular specimen experimentally shown to be nominally isotropic (The authors performed multiple tensile experiments for specimens extracted longitudinally and along several radial directions). \[d\epsilon_{ij}^{e}=\frac{1}{E}\big{\{}(1+v)d\sigma_{ij}-vd\sigma_{kk}\delta_{ ij}\big{\}} \tag{3}\]  
where \(d\sigma_{ij}\) represents an increment of the stress tensor, E is the elastic modulus, \(\upsilon\) is Poisson's ratio and \(\delta_{ij}\) is the Kronecker delta symbol. For plastic deformation, increments in plastic work/dissipation \((dW^{p})\) in terms of increments in equivalent plastic (\(d\bar{\epsilon}^{p}\)) strain are employed with an effective stress (\(\overline{\sigma}\)) and thus connect general multi-axial states to uniaxial experimental results in the following way;  
\[dW^{p}=\overline{\sigma}\cdot d\bar{\epsilon}^{p}=\sigma_{ij}\cdot d\epsilon _{ij}^{p} \tag{4}\]  
In Equations (1) and (4), the effective stress (\(\overline{\sigma}=\overline{\sigma}_{Ba}\)) employed in the Barlat yield function can be written;  
\[ \begin{array}{c}\overline{\sigma}=\overline{\sigma}_{Ba}=(\frac{1}{2})^{ \frac{1}{n}}(3\ J_{2}\ )^{\frac{1}{2}}(\left\{\left[2\cos\left(\frac{2\theta+\pi}{6}\right)\right]^{m }+\left[2\cos\left(\frac{2\theta-3\pi}{6}\right)\right]^{m}\right.\\ \left.+\left[-2\cos\left(\frac{2\theta+5\pi}{6}\right)\right]^{m}\right\})^{ \frac{1}{n}}\end{array}  \tag{5}\]  
where m is an integer with m = 8 for FCC materials such as aluminum alloys. The parameter \(\theta\) can be expressed as (The parameter, \(\theta\), in Equation (5) is unrelated to rotations associated with coordinate transformations in either plane stress or plane strain),  
\[\theta=cos^{-1}\left( \begin{array}{c}J_{3}\\ \frac{J_{2}}{2}\end{array} \right)\]where \(J_{2}\) and \(J_{3}\) are the second and third invariants of the deviatoric stress tensor, \(S_{ij}\). For the Barlat YId91 [17] anisotropic yield criterion, \(S_{ij}\), \(J_{2}\), and \(J_{3}\) are defined as follows:  
\[S_{ij}=\left[ \begin{array}{ccc}S_{x}&S_{xy}&S_{zx}\\ S_{xy}&S_{y}&S_{yz}\\ S_{zx}&S_{yz}&S_{z}\\ \end{array} \right]=\left[ \begin{array}{ccc}\frac{\mathrm{C}-B}{3}&H&G\\ H&\frac{\mathrm{A}-\mathrm{C}}{3}&F\\ G&F&\frac{\mathrm{B}-A}{3}\\ \end{array} \right]\]  
\[S_{ij}=\left[ \begin{array}{ccc}\frac{\mathrm{C}-bB}{3}&hH&gG\\ hH&\frac{\mathrm{a}A-\mathrm{C}}{3}&fF\\ gG&f&\frac{\mathrm{b}\bar{B}-aA}{3}\\ \end{array} \right]\]  
\[J_{2}=\frac{(fF)^{2}+(gC)^{2}+(hH)^{2}}{3}\]  
\[+\frac{(aA-\mathrm{C})^{2}+(cC-bB)^{2}+(bB-aA)^{2}}{54}\]  
\[J_{3}=\frac{(cC-bB)((AA-\mathrm{C})(bB-aA)}{54}\]  
\[+fghFGH-\frac{(cC-bB)(f)^{2}+(aA-\mathrm{C})(gC)^{2}+(bB-aA)(hH)^{2}}{6}\]  
In these equations, \(A=\sigma_{y}-\sigma_{z}\), \(B=\sigma_{z}-\sigma_{x}\), \(C=\sigma_{x}-\sigma_{y}\), \(F=\sigma_{yz}\), \(G=\sigma_{zx}\), \(H=\sigma_{xy}\).