Issue 37

G. Cricrì et alii, Frattura ed Integrità Strutturale, 37 (2016) 333-341; DOI: 10.3221/IGF-ESIS.37.44 334 In equation (1), I 1 is the first invariant of Cauchy stress tensor, J 2 is the second invariant of deviatoric tensor, K is a limit stress of base material, which takes account of the hardening effect due to plastic flow, and A , B and C are model parameters, function of variable density ρ . The simplest constitutive model combines a rule of plastic flow with an isotropic hardening law. Its basic hypothesis is to consider the powder grains only affected by plastic strains whilst their rearrangement is negligible. Kuhn, Green, Shima, Doraivelu, Fleck and many other authors [3-16] have presented material models characterized by a yield surface function of I 1 and J 2 , as described in (1). However, according to Biswas [17] these relationships, developed omitting grains rearrangement mechanism, are not suitable for modelling initial stages of compaction process, at lower density levels. Therefore, in this work a material model with two limit surfaces ( cap-cone model ) is presented in order to simulate all the die compaction steps of powders whose behaviour is both porous and granular. In such a way, it is possible to overcome the drawback highlighted by Biswas. A plastic flow law combined with friction mechanism is implemented in the finite element (FEM) commercial code ANSYS by means of the USERMAT subroutine written in Fortran. Moreover, several numerical simulations were performed to investigate the entire die compaction process, considering different friction coefficients between metal powders and die walls and punches. Friction parameters play a crucial role in the evaluation of loads applied to the die and consequently in the proper estimation of die strength and residual strains in the formed object Therefore, these analyses are helpful for a correct die design and final recognition of formed objects’ dimensions [18-20]. M ATERIAL MODEL he material model used in this study consists of two limit surfaces [21] F 1 and F 2 : the former, F 1 (σ), θ is related to the Mohr-Coulomb criterion ( cone surface ); the latter, F 2 (σ, σ c ), is related to a yield elliptical surface ( cap surface ) where σ c represents a hardening parameter, function of volumetric plastic strain. Such a formulation is suitable for modelling metal powders with extremely different relative density (from 0.2 r   to 1.0 r   ). The F 1 and F 2 limit surfaces are expressed in equations (2) and (3), respectively.   2 1 1 2 1 sin cos sin sin cos 0 3 3 J F I J c            if m c     (2)     2 2 2 2 2 , 0 tan c m c c c F J M                            if m c     (3) In these equations, I 1 is the first stress invariant, J 2 is the second invariant of the deviatoric stress tensor and σ m is the hydrostatic stress, defined as σ m = I 1 /3. The yield surface F 1 is obtained from the Mohr-Coulomb criterion, as a function of normal stress  n and shear stress   : 1 cos sin cos n F c         . (4) The parameters c and  are respectively the cohesion and the angle of internal friction of metal powder. The angle of internal friction  represents the angle of shearing resistance during compression, i.e. a measurement of resistance to relative sliding of grains. The cohesion c defines the shear strength of powder when no normal loads are applied to. Let the principal stresses be with magnitudes 1 2 3      , the equation for the surface (4) can be rewritten as (5).     1 1 3 1 3 1 1 sin cos 2 2 F c            (5) The principal stresses and invariants   1 2 , , I J  depend on the relationships: T