PSI - Issue 72

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

545

As a final note, upon examining these simulation results, it seems opportune to recall that Krajcinovic argued Krajcinovic (1996) and Krajcinovic et al. (1995), that mean field models are applicable if and only if the following three conditions are satisfied:













Pr

0,

max

 ,

,

(7)

L L

a

 L W rve

L





 



max

d

i

rve

x



where L d , L i , and L rve are the distance between microcracks, microcrack interaction range, and the linear size of the representative volume element, respectively;   x    is the volume average of micro-stresses; while ξ denotes the correlation length over which the fluctuations of the stress fields created by microcracks are correlated. Ray (2018) recently asserted that for heterogeneous materials “a theory based on a continuous coarse-grained description is untenable and a realistic computer simulation is almost unfeasible,” due to the complexities involved in the nucleation and growth of fractures, where defects interact cooperatively across a broad range of length scales. 4. Conclusion Over the past fifty years, theoretical studies on the fracture behavior of ferritic steels have extensively investigated the significant variability observed in cleavage fracture toughness under nominally identical experimental conditions within the ductile-brittle transition temperature range. In particular, statistical approaches based on Weibull-type distributions to characterize the scatter in fracture toughness are now well established. However, there is a risk that the various assumptions and approximations underlying these statistical models may fade from our memory. Although these models have been analytically developed and experimentally validated across a broad class of brittle materials—especially ferritic steels—they are occasionally regarded as definitive. On the other hand, it is important to emphasize that the derivations of these models involved a complex series of assumptions and mathematical formulations that warrant careful examination. The ongoing efforts based on particle dynamics-based CT simulations aim to critically investigate some of these foundational assumptions and approximations. These simulations may provide qualitative insights into the micromechanical evolution of damage and subsequent fracture in (quasi)brittle systems with varying degrees of quenched disorder. Damage snapshots and stress histories along the crack-tip ligament of pre-cracked specimens indicate a sequential fracture process. This observation, strictly speaking, contradicts the classical weakest-link hypothesis, which asserts that the failure of a single critical micro-element immediately initiates global rupture of the entire 3-D structural component. Such an immediate catastrophic collapse was neither observed nor is it likely to be observed within the present PD framework, regardless of the level of the material’s inherent disorder or brittleness. Instead, the simulations revealed that damage propagates progressively; influenced by loading conditions, stress state, and disorder level, ultimately culminating in catastrophic fracture. In essence, although the total accumulated damage prior to failure may be small, it nevertheless persists, evolves, and eventually leads to global failure. Acknowledgements This work was supported by the Ministry of Science, Technological Development, and Innovation of the Republic of Serbia. References Ruggieri, C., Leite, L., Ferreira, D., 2024, Statistical description of fracture toughness revisited: Implications for evaluation of the reference temperature, T0, and characteristic fracture toughness, Mechanics of Materials 196, 105055. https://doi.org/10.1016/j.mechmat.2024.105055 Djordjevic et al., 2023, Ductile-to-brittle transition of ferritic steels: A historical sketch and some recent trends, Engineering Fracture Mechanics 293: 109716. https://doi.org/10.1016/j.engfracmech.2023.109716 Mastilovic S., 2022, Introduction to Computational Mechanics of Discontinua, in: "Mathematical Physics: Volume II - Numerical Methods", In: Sedmak, A., Kuzmanovic, D. (Ed.), ESIS (European Structural Integrity Society) Publishing House, 213-285, ISBN 9788831482561 Krajcinovic, D., 1996, Damage Mechanics, Elsevier. ISBN 0-444-82349-2.