The fracture of ductile materials is often the result of the nucleation, growth and coalescence of microscopic voids. In the present talk, we mainly focus on the growth process of voids. In dynamic loading, micro-voids sustain an extremely rapid expansion which generates strong acceleration of particles in the vicinity of cavities. These micro-inertia effects are playing a significant role in the development of damage and can be accounted for in the macroscopic response of the porous medium via a multi-scale approach proposed in [1]. As a consequence, the overall macroscopic stress is the sum of two contributions: a static term (micro-inertia independent term) and a dynamic term (micro-inertia dependent term).

In this work, we focus on the material response of porous medium with cylindrical voids. As a consequence, a cylindrical representative volume element (porosity f) for the porous material is considered, similar to the one proposed by Gurson [2]. The radius of the void is a, the length is L. The outer radius of the RVE is b. The static term is derived from the yield function proposed by Gurson [2]. The dynamic stress is evaluated analytically from the trial velocity field of Gurson [2] valid for cylindrical geometry. It is shown, that as for spherical configuration, the dynamic stress is scaled by the mass density, the size of the voids, the porosity, the macroscopic strain rate tensor and the time derivative of the strain rate tensor. An important outcome of the model is the differential lengthscale effect which exists between in- plane and out of plane components of the macrostress. Namely, it is observed for axisymmetric loading that in-plane dynamic stress components are only related to the radius a while the out of plane stress component is linked to the radius a and the length of the RVE, L.

In this talk, we present the mechanical response of the medium when the RVE is subjected to spherical loading or axisymmetric plane strain loadings (with D33=0 and D33=constant) in conjunction with in plane isotropic stress loading (at constant stress rate). While for plane strain loading in quasi static condition, the overall axial stress is half of the sum of in-plane component, in dynamic conditions, the inertia contribution reveals a difference. An important result of the proposed theory is the length effect of the RVE, which does not exist in quasi static conditions since the quasi static porous response is related to the porosity. The analytical model is validated based on comparisons with finite element calculations (Abaqus/Explicit).

Acknowledgements : The research conducted in this work has received funding from the European Union's Horizon 2020 Programme (Excellent Science, Marie-Sklodowska-Curie Actions) under REA grant agreement 675602 (OUTCOME Project).

References :

[1] A. Molinari, S. Mercier, (2001). Micro-mechanical modeling of porous materials under dynamic loading. J. Mech. Phys. Solids, 49, 1497-1516.

[2] A. L. Gurson, (1977). Continuum theory of ductile rupture by void nucleation and growth : Part I - yield criteria and flow rules for porous ductile media. J. Eng. Mater. Technol., 99, 2–15.

[3] C.Sartori, S.Mercier, N.Jacques, A. Molinari (2014). Constitutive behavior of porous ductile materials accounting for micro-inertia and void shape. Mechanics of Materials, 324-339.