Calibration of Gurson-type models for porous sheet metals with anisotropic non-quadratic plasticity

Abstract

The growth and coalescence of voids in sheet metals are not only the main active mechanisms in the final stages of fracture in a necking band, but they also contribute to the forming limits via changes in the normal directions to the yield surface. A widely accepted method to include void effects is the development of a Gurson-type model for the appropriate yield criterion, based on an approximate limit analysis of a unit cell containing a single spherical, spheroidal or ellipsoidal void. We have recently [2] obtained dissipation functions and Gurson-type models for porous sheet metals with ellipsoidal voids and anisotropic non-quadratic plasticity, including yield criteria based on linear transformations (Yld91 and Yld2004-18p) and a pure plane stress yield criteria (BBC2005). These Gurson-type models contain several parameters that depend on the void and cell geometries and on the selected yield criterion. Best results are obtained when these key parameters are calibrated via numerical simulations using the same unit cell and a few representative loading conditions. The single most important such loading condition corresponds to a pure hydrostatic macroscopic stress (pure pressure) and the corresponding velocity field found during the solution of the limit analysis problem describes the expansion of the cavity.However, for the case of sheet metals, the condition of plane stress precludes macroscopic stresses with large triaxiality or ratio of mean stress to equivalent stress, including the pure hydrostatic case. Also, pure plane stress yield criteria like BBC2005 must first be extended to 3D stresses before attempting to develop a Gurson-type model and such extensions are purely phenomenological with no due account for the out- of-plane anisotropic properties of the sheet. Therefore, we propose a new calibration method for Gurson- type models that uses only boundary conditions compatible with the plane stress requirement. For each such boundary condition we use a spectral method to solve the limit analysis problem, using as spectral basis a newly developed Mie decomposition of incompressible velocity fields for ellipsoidal cells with confocal ellipsoidal voids, extending the well-known Lee and Mear family corresponding to the spheroidal axisymmetric case [1]. We thus obtain a series of points located on the Gurson-type yield surface and their corresponding normal directions. These points are used as input to some enhanced parameter identification method developed for anisotropic yield criteria [1] to determine the calibration of the Gurson-type models. We finally use these newly calibrated Gurson-type models to evaluate forming limits for porous sheet metals.