PSI - Issue 45

56 Nhan T. Nguyen et al. / Procedia Structural Integrity 45 (2023) 52–59 Nguyen et al. / Structural Integrity Procedia 00 (2019) 000 – 000 5 In the above expressions (Eqs. (10-13)), =− x 3 is the mean stress under geotechnical sign convention (compression positive); =√ 3 2 ( x − 1 3 x )( x − 1 3 x ) is the deviatoric shear stress; p x is the plastic strain; is the breakage energy threshold; is the ratio between the deviatoric shear and mean stresses at the critical state; ̇ is the non-negative plastic multiplier and is the plastic-breakage coupling angle. 3. Results and analyses This section examines the capacity of the proposed model in the post-localisation analysis compared with the drained triaxial test on Berea sandstone for a cylindrical sample, with a diameter of 18.4 mm and length of 38.1 mm (Baud et al., 2004; Baud et al., 2006). A new set of parameters with more rigorous calibration, compared to our previous work in Nguyen et al. (2021b), using the elastic modulus, , and Poisson ratio, , was used: = 13 GPa; ν = 0.25; =0.6; = 380 MPa; = 1.46; = 80°; = ℎ =0.3 . The dual-scale enrichment involves the sample characteristic length, , which was obtained from the ratio of the specimen volume and the surface area of a pure compaction band assuming horizontally oriented. Due to the lack of information on the band thickness, ℎ , for Berea sandstone, its value was assumed based on the image of damaged specimens made of Bentheim sandstone of which ℎ ≈ 18 , provided in the experimental results (Baud et al., 2004) (Fig. 2). Although it is possible to capture the evolving thickness ℎ of the band (Nguyen and Bui, 2020) which reflects the progressive nature of compaction localisation, the value of ℎ in this work was kept constant for straightforward comparison with the experimental data. a b c d

Fig. 2. The propagation of compaction band, represented by the dark lines, in Bentheim sandstone undergoing triaxial shearing at 300 MPa confining pressure after different levels of axial strain: (a) 1.4%; (b) 3.1%; (c) 4%; and (d) 6% (Baud et al., 2004).

Following the obtained set of parameters, the bifurcation analysis (Nguyen et al., 2021b) produced the range of localisation activation in the triaxial stress space with two stress paths shearing at 150 MPa and 200 MPa, see Fig. 3.

Fig. 3. Bifurcation zone of Berea sandstone based on the breakage model.

The post-localisation analysis involves two settings of the dual-scale framework including the outside bulk being enforced to behave elastically, or allowing an inelastic response to occur. The latter configuration is an improvement to the former one as an attempt to capture the propagation of the compaction band as observed in Fig.2, where another band will develop outside of the localised zone and then propagates all across the volume until fully compacted. In Fig. 4, the macro response from the model was compared with the experimental response with the elastic outside setting which implies the validity of the calibrated set of parameters and the model performance. The stress paths triaxial space for the macro, inside and outside behaviour are illustrated in the correspondence with the initial breakage yield surface with the slope : = 3: 1 (Fig. 5). Once the stress state reaches the initial yield surface, localisation occurs and the three responses follow different paths. The convergence of inelastic responses