In his famous paper, Gurson,1977 proposed an upper bound limit analysis approach of a hollow sphere and a hollow cylinder having a von Mises solid matrix. Several extensions of Gurson model have been further developed in literature. Considering the micromechanical modeling for applications to cohesive geomaterials, Guo et al., 2008 and Cheng et al., 2015 have both adopted the theoretical and numerical homogenization to take into account the plastic compressibility of the matrix respectively obeying the associated and non-associated Drucker-Prager yielding laws. However, the aforementioned models have been derived by assuming that the solid matrix obeys a yield criterion either depending on the second stress invariant (i.e. J2 plastic rule) or accounting for the first and the second ones. Nevertheless, the solid matrix of some engineering-relevant porous materials may exhibit a more complex plastic behavior that also depends on the third stress invariant, that is on the stress-Lode-angle. Few attempts have been made in literature to include the influence of all the three isotropic stress invariants for describing strength properties of porous media. Mention can be made of the studies by Lemarchand et al., 2015, Anoukou et al., 2016, Pastor et al., 2016. Some of these studied have been devoted to the special case of Mohr-Coulomb plastic matrix.

We propose a numerical estimate of the macroscopic strength for ductile porous material by considering the local plastic behavior as dependent on all the three isotropic stress invariants and by referring to the case of axisymmetric strain-rate boundary conditions. In this light, the general and flexible yield criterion proposed by Bigoni et al., 2004 and recently particularized by Brach et al., 2018 is considered as a promising candidate to comply with benchmarking indications on local strength properties. This allows to effectively describe the limit behavior of a broad class of materials, such as pressure independent metals, high strength shape memory alloys, pressure sensitive geomaterials and polymers, quasi-brittle concretes, etc. Specifically, a Finite Element -based limit analysis procedure is implemented in order to compute the macroscopic yield surfaces. This allows to assess theoretical predictions and to compare to available numerical upper and lower bounds especially by paying particular attention to the matrix Lode angle effects.

References:

Anoukou, K., Pastor, F., Dufrenoy, P., Kondo, D., 2016. Limit analysis and homogenization of porous materials with Mohr–Coulomb matrix. Part I: theoretical for- mulation. J. Mech. Phys. Solids 91, 145–171.

Bigoni, D., Piccolroaz, A., 2004. Yield criteria for quasibrittle and frictional materials. Int. J. Solids Struct. 41, 2855-2878.

Brach, S., Dormieux, L., Kondo, D., Vairo G., 2018. Nanoporous materials with a general isotropic plastic matrix: Exact limit state under isotropic loadings. International Journal of Plasticity, 89, 1-28.

Cheng, L., Jia. Y., Oueslati, A., de Saxcé, G., Kondo, D., 2015. A bipotential-based limit analysis and homogenization of ductile porous materials with non-associated Drucker–Prager matrix. J. Mech. Phys. Solids, 77, 1-26.

Gurson, A.L., 1977. Continuum theory of ductile rupture by void nucleation and growth. J. Engng. Mater.

Guo, T.F., Faleskog, J., Shih, C.F., 2008. Continuum modeling of a porous solid with pressure-sensitive dilatant matrix. J. Mech. Phys. Solids 56, 2--15.

Lemarchand, E., Dormieux, L., Kondo, D., 2015 Lode's angle effect on the definition of the strength criterion of porous media. International Journal for Numerical and Analytical Methods in Geomechanics, Wiley, 2015, 39 (14), pp.1506--1526.

Pastor, J.F., Anoukou, K., Pastor, J., Kondo, D., 2016. Limit analysis and homogenization of porous materials with Mohr-Coulomb matrix. Part II: numerical bounds and assessment of the theoretical model. J. Mech. Phys. Solids 91, 14–27.