13th International Conference on Fracture June 16–21, 2013, Beijing, China -4- 3. Modeling of deformation and failure behavior of dissimilar spot welded joints 3.1. Material model The constitutive model developed by Gologanu and Leblond [12] for porous media is an extension of the Gurson model [13] which takes into account the effect of void shape. In addition to the porosity f, the void shape S or w is considered in the flow potential of the model. Considering an oblate or prolate axisymmetric spheroidal void, shown in Fig. (4), with the semi-axes a along the x2- (axial direction) and b along the x1- and x3-direction (radial direction), w and the S are defined as w=a/b and S=ln(a/b)=ln(w). Spherical voids are characterized by S = 0 and w = 1 respectively. The shape parameters for prolate voids are S > 0 and w > 1, while S < 0 and w < 1 is true for oblate voids. Figure 4. Spheroidal axisymmetric prolate (left) and oblate (right) void in a representative volume element with the shape of a confocal spheroid with the semi-axis A along x2 and B along x1 and x3 The porosity of the representative volume element is given by 2 2 AB ab f = . (2) The functional form of the flow potential, which was derived by Gologanu using an approximate load limit-analysis, is given in Eq. (3). 2 2 M ij T C σ Φ= + ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ Σ + + M h q g g f σ κ )cosh 2 ( 1)( 1 − 2 ( 1) +g − ( ) 0 2 2 1 + = q g f (3) with ij ij ij T T T 2 3 = (4) Tij and Σh are anisotropic generalizations of the deviatoric and the hydrostatic stresses. σM is the equivalent yield stress of the matrix material. The parameters C, q1, g and κ are functions of the void volume fraction f and the void shape S. The evolution of the void shape S&and the porosity in terms of the rate of void growth growth f & and nucleation nuc f & are given in Eq. (5) and Eq. (6) – (8). p b kk p a a S S S ε ε &= & + & ′ (5) The parameters Sa and Sb are functions of the void shape and the stress triaxility. p a ′ ε& and p kkε& are the deviatoric part of the plastic strain rate in the direction of the semi-axis a and the rate of
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