PSI - Issue 45

Nhan T. Nguyen et al. / Procedia Structural Integrity 45 (2023) 52–59 Nguyen et al. / Structural Integrity Procedia 00 (2019) 000 – 000

55

4

̇

= ̇

(6)

while they can be used to homogenize the rate of macro stress ̇ as follows: ̇ = (1 − ) ̇ + ̇

(7)

The derivation of conditions in Eqs. (6) and (7) can be found in detail in (Nguyen et al., 2012; Nguyen et al., 2014; Nguyen et al., 2016). The macro constitutive relationship can be obtained as follows (see details on the derivation in the mentioned papers): ̇ = ̇ = [(1 − ) + − (1ℎ− ) −1 ( − ) ( − )] ̇ (8) in which denotes the macroscopic tangent stiffness, and = ℎ o + 1− ℎ i . This takes into account the interaction between behaviour inside the localisation band and the surrounding bulk along with the anisotropy due to the inclined localisation band. Consequently, the inhomogeneous and size-dependent responses in the post-localisation can be naturally addressed by our model. Eq. (8) can be reduced to the form of constitutive relationship in the traditional continuum model if =1 , indicating the ability of this approach in handling the transition between diffuse and localised regimes. 2.2. The base constitutive model accounting for grain-crushing mechanisms In this model, the constitutive relationship of the inelastic behaviour in the pre-localisation stage ( ̇ h = h ̇ h with ̇ h = ̇ and ̇ h = ̇ for the homogeneous behaviour) or inside the localisation bands in the post-localisation stage ( ̇ i = i ̇ i ) is based on the theory of breakage mechanics (Einav, 2007a,b) which tracks the evolution of grain size distribution to represent the grain crushing process. In this sense, h (used in Eq. (1)) and i (used in Eqs. (5) and (8)) can be specified in the below form: = (1 − ) − [(1 − ) + 1− ] (1− ) (1− ) −( − 1− ) (9) where is the grading index, while , , , y and are stress, breakage energy and grain-crushing internal variables, respectively, of the nonlinear behaviour in the homogeneous stage. Here, “x” either stands for “h” regarding homogeneous or “i” regarding the region inside localisation. The above form of tangent stiffness can be derived from the following list of key formulations of the breakage model (Einav, 2007a,b), including the constitutive relationship, yield function, and the flow rules for plastic strain and evolving breakage, respectively: = (1 − ) ( ̇ − ̇ ) (10) = 2 [ ( ) 2 + ( ) 2 3 ]( 1− 1− ) 2 +( ) 2 −1≤0 (11) ̇ = ̇ = ̇ [− 2(1− ) 2 2 3 + 3( − 1 3 ) ( ) 2 ] (12) ̇ = ̇ =2 ̇ (1− ) 2 2 (13)