PSI - Issue 72

Sreten Mastilovic / Procedia Structural Integrity 72 (2025) 538–546

541

The coordination number z = 6 and the interparticle distances rij characterize the randomness in the system’s morphology. Specifically, this PD model incorporates both geometric and structural disorder: the initial distances between particles particles i and j (r0ij) and the corresponding bond stiffness (kij) are randomly sampled from normal and uniform distributions, respectively, within the specific ranges   r r r r ij r      2 0 and   k k k k ij k      2 . The geometric-order parameter λr and the structural-order parameter λk , each ranging from 0 to 1, control the degree of disorder in geometry and stiffness—where 1 represents perfect order. Additionally, random failure thresholds are defined by the critical bond elongation,  ij = (rij/r0ij – 1) =  cij, which is sampled from the uniform distribution within the range   c c c 2           ij with the ruptural-order parameter (0 ≤ λε ≤ 1). PD model parameters denoted with an overline ( c  r,k , ) represent nondimensional average values, which can, in principle, be determined from the material properties under study Mastilovic (2022), Krajcinovic (1996) and Mastilovic (2008). 2.1. PD-CT model PD model of the pre-cracked specimen used in this study (mimicking the CT specimen and dubbed PD-CT) is shown in Fig. 1. The displacement-controlled quasistatic loading is applied by the way of two pins (Δ/2 each) to mimic the standard fracture mechanics test from ASTM E1921 (2022). The four virtual measurement gages, marked A to D, located equidistantly along the specimen ligament, are used to calculate stress components within their respective areas, Ω, in accordance to well-established MD practice, Mastilovic (2022), Allen et al (1987):           ) (1,..., , , (1, 2) , , 2 1 2 1 , , i j k N r r r f r r r dr d ik ik ik k i i k ik ik ik ik k i i k ik j                    (6) (Note that the lower-case alphabetic indices ( i, j ) are reserved for numbering of particles, while the lower-case Greek letter subscripts ( α, β ) refer to tensor components. For instance, f ij designates the force between particles i and j with components ( f ij ) α , ( α= 1, 2), ( i, j = 1,..., N ).) The displacement step used in simulations, δ Δ= 2·10 -6 · a 0 , is normalized with respect to the initial crack length a 0 = W /2 (Fig. 1) to to enable objective comparison of results across different specimen sizes ( W ) in future studies.

Fig. 1. PD-CT specimen W × H = 180×170. (a) Displacement-controlled loading applied through two loading pins, with four virtual measurement gages (A-D) positioned equidistantly along the crack plane across the ligament length W - a 0 . (b) Examples of particle lattice geometric disorder ahead of the sharp crack tip: strong disorder shown at the top and weak disorder at the bottom. Note that all lengths are given in the nondimensional form: in the unit of the model resolution length, l ch , identified in this study with the average crystal-grain diameter of the material. The geometrical and structural disorder of the PD-CT model used in this study as a basic case is defined by the following parameter set: ( λ r , λ k , λ ε ) = (0.02, 0.9, 2/3), while the average model parameters are 1  r , 50  k , and 0015 c .   . (Conversion of the