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

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criterion (Hill, 1963; Rudnicki and Rice, 1975; Rice, 1976; Rice and Rudnicki, 1980; Chambon et al., 2000) assumes that the loss of positiveness of the acoustic tensor is the trigger for the onset of localisation: ⃗ [ ( )]≤0 (1) where = ℎ is the localisation acoustic tensor; ℎ is the homogeneous tangent stiffness tensor derived from the classical breakage model (Einav, 2007a,b), see further details in sub-section 2.2 and Nguyen et al. (2021a) in the pre-localisation stage and is the normal vector representing the orientation of the localisation band obtained from the bifurcation criterion in Eq. (1) In the post-localisation stage, localised failures induce discontinuous deformation across the band boundary where this phenomenon can be described by two separate strain rates outside ( o ) and inside ( i ) localisation regions in the following forms: ̇ = ̇ − ℎ ( ̇ ) (2) ̇ = ̇ + 1− ℎ ( ̇ ) (3) The above expressions are obtained through the use of kinematic enrichment (Neilsen and Schreyer, 1993; Oliver, 1996; Borja, 2000) based on the macro strain rate, , the relative velocity, ̇ , between two faces of the localisation bands, and parameters representing properties of these bands including thickness, ℎ , and orientation, , along with their interaction with the size of the volume element/specimen including volume fraction = ℎ with = being the characteristic length of the volume element, is the cross-section area presented in blue, while and are the volume inside and outside of the localisation zone, respectively, and = + is the total volume (Fig. 1). Fig. 1. The concept of dual-scale framework describing an element embedded with a localisation band Given local mesoscale strain increments ( ε̇ kl , ε̇ kl ) in Eqs. (2-3), stress increments outside ( σ̇ ij ) and inside ( σ̇ ij ) the localisation bands can be determined through two following separate constitutive relationships: ̇ = ̇ = [ ̇ − ℎ ( ̇ ) ] (4) ̇ = ̇ = [ ̇ + 1− ℎ ( ̇ ) ] (5) where o and i are stiffness tensors outside and inside the compaction bands, respectively. It is noted that o is equal to the constant elastic stiffness tensor = ( + )+( − 2 3 ) , where = 2(1+ ) denoting the shear moduli, = 3(1−2 ) denoting the bulk moduli and being the Kronecker delta tensor, to reflect the reversible behaviour of the surrounding bulk. i is formulated based on the breakage model (Einav, 2007ab, see sub-section 2.2) of stresses inside the localisation zones for capturing the irreversible deformation governed by the interaction between processes of grain rearrangement and crushing, as described in sub-section 2.2. These two stress rates must meet the traction equilibrium condition across the boundary of compaction bands (Nguyen and Bui, 2020) in the following form: