On the overall yield function of a hollow sphere

  Motivated mostly by analytical (or formal) considerations, a vast majority of the works dedicated to the study of the overall plastic response of a porous metal assumes that the sound material (or matrix) obeys a quadratic yielding criterion (von Mises or Hill'48). However, most metals of technological interest today feature significant deviations from the quadratic form of yielding. Therefore a realistic modeling of their behavior, while sustaining damage,  requires appropriate consideration of their yielding properties.
  The present contribution applies Gurson's methodology to the estimation of the overall yield function of a porous rigid-plastic representative volume element (RVE) who's matrix obeys a non-quadratic yielding criterion. Analytical difficulties that have limited in the past the range of applications of this methodology are related to the (explicit) calculation of the local dissipation and corresponding averaging formulas. These are circumvented here by employing a general (Fourier) representation for the local dissipation (corresponding to an arbitrary yielding criterion) and by calculating numerically the averaging integrals. Applications are shown for the "classical" RVE in the shape of a hollow sphere. First, the case of the von Mises matrix is investigated (mainly for validation of the new approach; however, even in this case new results are reported, since our calculations are "exact" by comparison with previous approaches based on upper-bound estimations) and then the case of a matrix obeying the Hershey-Hosford criterion.