13th International Conference on Fracture June 16–21, 2013, Beijing, China -3- The void nucleation intensity, Ais a function of p mε the equivalent plastic strain in the matrix material, and is assumed to follow a normal distribution as suggested by Chu and Needleman (1980) [6]: 2 1 exp 2 2 p N m N N N f A s s ε ε π ⎡ ⎤ ⎛ − ⎞ ⎢ ⎥ = − ⎜ ⎟ ⎢ ⎥ ⎝ ⎠ ⎣ ⎦ , (8) where Nf is determined so that the total void volume nucleated is consistent with the volume fraction of particles; Nε is the mean equivalent plastic strain for void nucleation; and Ns is the standard deviation of the distribution. 2.2. Void coalescence criterion 2.2.1 Plastic limit load criterion Thomason (1985, 1998) [8] found that the localized deformation mode by intervoid matrix necking can be characterized by a plastic limit load which is not fixed but is strongly dependent on the void geometry and stress states. The condition for void coalescence can be written as: 1 1 L σ σ = , (9) where 1 Lσ represents the capacity of the material to resist void coalescence; and 1σis the maximum principal stress at current yield surface of a material point. Using a 3D unit cell containing an axisymmetric ellipsoidal void, Thomason acquired the plastic limit load to void coalescence with the following form 2 1/2 2 1 2 1 L x x z x R R R X R X X π σ α β σ − − ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎜ ⎟ = + − ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ − ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ , (10) where xR , zR and X are the current radii of the ellipsoidal void in the x- and z- axes and the current length of the cell in the x-axis, respectively; the local coordinates system is constructed so that x-, y- and z- axes represent the minor, medium and maximum principal stress directions. αand βare constants which are suggested as 0.1 and 1.2 by Thomason. Pardeon and Hutchinson (2000) [7] conducted a large number of cells calculations and found the dependence of αand βon hardening exponent n. Their simulation results showed that βis almost a constant and can be taken as 1.24 while ( ) 2 0.1 0.217 4.83 (0 0.3) n n n n α = + + ≤ ≤ , (11) If the void is assumed to be always spherical, the void/matrix geometry in Eq. 10 can be directly determined from the current void volume fraction f and current principal strain 1ε, 2ε, 3ε by following equations (Zhang, 2001)[13]: 1 2 3 3 3 4 x y z f R R R eε ε ε π + + = = = , (12) 1 2 / 2 X Y eε ε+ = = , (13) 2.2.2 Equivalent plastic strain criterion By assuming the existence of a periodic distribution of voids, the material can be considered as an array of cubic blocks with each block being a unit cell having a void at its center. Failure of the unit cell occurs when localization of plastic flow takes place in the ligament (Koplik and Needleman,
RkJQdWJsaXNoZXIy MjM0NDE=