Constitutive equations for porous solids with matrix behaviour dependent on the second and third stress invariants

- We introduce consistently Lode angle in Gurson's analysis both for the yielding of the matrix and effects arising from stress heterogeneity.
- We give a general parametric representation for the effective yield domain of the porous material.
- We give some explicit closed form results for particular loadings (pure shear and hydrostatic loadings).
- We furnish a semi-explicit equation of this yield domain.
- We furnish various illustrations of the results.

Constitutive equations are developed for voided materials and ductile fracture taking into account possible effects of Lode angle in the yielding behaviour of the matrix. The Gurson criterion (Gurson, 1977) [4] is generalized to such circumstances. A semi-closed form expression , similar to the Gurson criterion is obtained for the effective yield criterion for the porous solid and involves four different functions , all dependent on the macroscopic stress triaxiality and Lode angle but are not generally available in closed form. In parallel, a parametric representation of the effective yield criterion is provided which allows for the derivation of closed form results for pure shear stress states and also at very high stress triaxialities. In the former case corresponding to a zero macroscopic mean stress, the contour of the yield domain in the π-plane has exactly the shape of the yield surface of the matrix in the deviatoric plane but a size reduced by a factor 1−f, with f the porosity of the voided material. In the latter, effective yield stresses for the voided material are slightly different from the Gurson result and found to be set by the yield stress at a microscopic stress Lode angle π3 for very high positive triaxiality and by the yield stress at a microscopic stress Lode angle 0 for very high negative triaxiality. Various numerical results are furnished to illustrate all the obtained results.