13th International Conference on Fracture June 16–21, 2013, Beijing, China -3- MGs to be 0 . The corresponding shear strength 0 satisfies 3 1 cosh3 2 0 0 y m y .Therefore, the new yield criterion can be obtained from Eq.(1) as 0 0 2 sech 3 2 1 y m m J , (2) where 0 f is the free volume increment. It is obvious that the shear strength is softening with increasing free volume concentration. Following the self-consistent dynamic free volume model [23], the local time rate of change of the free-volume concentration is p e e p e f G Λ R , (3) where is a material parameter of order unity, R is a free-volume creation function defining the free volume produced by a unit shear strain and is given by ss G ( ss is the effective stress at steady state) [29], 2 3 p ij p ij p e e e ( 3 ij p kk p ij p ij e ) is the effective plastic shear strain rate, eis the effective stress, and G is the shear modulus at room temperature. Considering the shear dilatancy inherent in the deformation of MGs, we introduce the dilatancy factor and invoke Q as the plastic potential as below 0 0 2 sech 3 2 1 y m m Λ Q J . (4) When , the deformation is non-associative. The deviatoric components of the plastic deformation strain tensor are obtained as kl kl ij p ij Q H e 1 , (5) where H is the plastic hardening modulus. If we adopt the spineless strain rate, the generalized constitutive relation is recast as ij kk ij kk kl kl kk kl kl ij ij ij K J s H J s H J s G s D 9 3 ' 3 2 ' 3 ' 2 2 2 2 2 2 , (6) where ki jk kj ik ij ij is the Jaumann rate of the true stress ij , ' 3 sinh 3 2 2 f m y , ' 3 sinh 3 2 2 f m y , and ss y m J GΛ H 1 3 1 3 sech 3 2 2 0 0 . The instantaneous rate of the deformation and the spin tensor are respectively 2 i j j i ij D v x v x , and 2 i j j i ij v x v x .The stress rate and deformation rate are related by kl ijkl ij L D , (7) where ijkl L is the elasto-plastic modulus tensor.
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