PSI - Issue 68

Nhan T. Nguyen et al. / Procedia Structural Integrity 68 (2025) 91–98 N.T. Nguyen et al. / Structural Integrity Procedia 00 (2025) 000–000

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These dissipation ratios are implemented into the model as input parameters. Their correspondence to the evolution of inelastic internal variables is the essence of the formulation to dictate how much dissipation each mechanism (damage, volumetric, shear) contributes to the work produced from the constitutive model. 2.2. Rate-dependent enhancement In order to capture the rate-dependent behaviour of rocks, this study employs the Perzyna viscoplastic enhancement (Perzyna, 1966) within the proposed damage-plasticity model. In rate-sensitive materials, plastic deformation does not occur instantaneously once the yield criterion is met; instead, it evolves over time, dependent on the degree to which stress exceeds a defined yield surface. The Perzyna enhancement addresses this by allowing plastic strain to accumulate when the yield function exceeds a critical value, effectively introducing a "viscous" delay in plastic response that increases with the rate of applied loading. The Perzyna viscoplastic framework modifies the plastic flow rule to account for the overstress, which is the amount by which the applied stress exceeds the yield criterion. The incremental plastic multiplier, defined by its rate form over a specific time increment, is determined as follows: = ̇ = ⟨P⟩ 0 S (17) This enhancement leads to the rate-dependent flow rules as: = ̇ = (P ∗ (, $ = ⟨P⟩ 0 S (P ∗ (, $ (18) T $ U = ̇ T $ U = (P ∗ (, 12 = ⟨P⟩ 0 S d (P ∗ (, ! (, ! (, 12 + (P ∗ (, # (, # (, 12 e (19) where is the viscosity parameter with unit / , and is a dimensionless power-law exponent controlling the rate sensitivity; T $ U is the total plastic strain increment tensor while the generalised dissipative stress tensor VW can be considered equivalent to true stress VW ; the Macaulay operation applied on the overstress yield value is defined as 〈 〉 = if > 0 for inelastic state, and 〈 〉 = 0 if ≤ 0 for the elastic state; and is the time step that is effective for the increment damage/plastic strain. This viscoplastic framework implies that a lower value results in faster strain rates and increases the rate-dependency in the material response. 3. Strain rate inside FPZ from simulation of three-point bending test The following parameter set of quasi-brittle material was used throughout all numerical results provided in this paper: Young’s modulus = 45 GPa, Poisson’s ratio = 0.24 , uniaxial tensile strength O =2.9 MPa; the width of crack band ℎ=8 mm, the fracture energy @ =0.04717 MPa.mm, and three dissipation ratios + =0.4, # $ = −0.3, & $ =0.9 . Aside from the fundamental material parameters like , , O and @ which were obtained from the physical properties of the tested material, the model parameters of the couple plasticity-damage scheme were calibrated with the uniaxial tension and three-point bending response. Further details can be found in Nguyen et al. (2024). The performance of the model under uniaxial tension for rate-independent response was shown in Fig. 1a where variation of the constitutive behaviour with respect to change of dissipation mechanism was highlighted. It was observed that by allowing damage to dominate the dissipation process instead of frictional plastic strain, the material would show a more brittle response. Then, the rate-dependent enhancement of the proposed damage model was calibrated such that the viscosity parameter was modified to ensure the material behaviour predicted by the rate dependent model matches the rate-independent response in the quasi-static range, as shown in Fig. 1b. The calibrated value, in this case, was = 10 /X ( ⁄ ) , which was then used to assess the capability of the rate-dependent enhancement. In Fig. 1c, the constitutive response under uniaxial tension was driven with various axial strain rates where the peak strength would reduce for a slow loading rate.