## For the proposed set of X W constitutive damage

In Fig. 2 the effects of Lode angle on the yield surface for different type of fracture models are shown. Fig. 2-a displays the stress triaxiality fracture model in which the Lode angle effect is ignored. So, the fracture loci are circles, which are of the same shape as the von Mises yield locus. Fig. 2-b displays the maximum stress fracture models such as Brozzo et al. [45] model. The fracture loci in this XMD8-92 category are hexagon, which are the same shape as the Tresca locus. In Fig. 2-c fracture locus on the octahedral plane for Wilkins et al. model [34] that uses a separate parameter, i.e. the stress asymmetry A to quantify the Lode angle dependence is shown. As demonstrated in mantle figure, the circular shape of the von Mises yield criterion is improved to a petal like shape. In Fig. 2-d fracture locus on the octahedral plane for (X-W) model [20] that uses Eq. (17) to quantify the Lode angle dependence is shown [20] and [34]. As seen, symmetric conditions of Lode angle dependence function on the octahedral plane for Wilkins et al. and (X-W) models are identical.