13th International Conference on Fracture June 16–21, 2013, Beijing, China -4- 2.2 Constitutive instability analysis We ascribe the initiation of shear band in MGs to bifurcation arisen from the constitutive description of the homogeneous deformation. The shear bands inclination is obtained at the onset of instability. An initially uniform deformation field of a homogeneous MG plate is considered, see Fig. 1. The constitutive instability is viewed as a formation of a narrow localized band of deformation within the plate under external load [17]. Use rectangular Cartesian coordinates ( 1x , 2x , 3x ), such that the 2x -direction is normal to the planes bounding the band. Outside the band the velocity field remains uniform and within it varies only in the direction normal to the band. Thus, the non-uniformities in the rate of deformation field are expressed as , ( , ) 1, 2,3 2 2 i j v x g x j i j i , (8) where iv is a velocity component, denotes the difference between the local field inside the band and the uniform field outside, and the functions i g of 2x are nonzero only within the band. Fig.1 Illustration of shear band localization. The condition 0 i ij x and 0 i ij t x make sure that the stress equilibrium continues to be satisfied at the inception of bifurcation. The stress rates at incipient localization from the original uniform field is thus following 0 i ij x , (9) where the superposed dot denotes its material time rate. The condition of the continuity of the stress rate at the band interfaces can be expressed as 0 2 j , 1, 2,3 j . (10) Since ij is not invariant under rigid rotations, by introducing the Jaumann (co-rotational) stress rate pi jp pj ip ij ij . Eq. (10) can be regarded as a set of three quasi-homogeneous equations in 1g , 2g , 3g and the conditions for bifurcation are merely those for which solutions other than 0 3 2 1 g g g exist. If at the bifurcation of deformation-rates, the values ijkl L remain the same inside and outside the band, the following difference can be formed based on Eq. (7): k ijk kl ijkl ij L D L g2 . (11) Combining Eqs. (10) and (11) yields a set of linear, homogeneous equations in g’ s ,
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