PSI - Issue 39

Bineet Kumar et al. / Procedia Structural Integrity 39 (2022) 222–228 B. Kumar et al./ Structural Integrity Procedia 00 (2019) 000–000

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scales, and each scale has some independent fracture mechanism, which can be found in the literature, (Le and Bažant (2011), Le, Bažant and Bazant (2011)). The accumulation of Representative volume elements (RVEs) from different scales in the macro crack formation eventually leads to a size effect for small and medium-sized structures, but its effect gets negligible comparatively and shows brittle behavior for a large structures, (Bazant and Kazemi (1990)). Therefore, it is essential to understand the scale effect in crack path behavior and its introduction into the crack growth rate expression to avoid the size effect in the prediction of life of structure under fatigue loading conditions. This paper aims to propose an analytical formulation for fatigue crack growth rate by considering the hierarchical fracture stages in its crack path inside the cyclic fracture process zone. Random jumps of atoms over activation energy barriers in the process of chemical bond breakage between CSH gels have been assumed in the mode I cyclic loading condition. Scale effects have been considered to encounter the specimen size effect. Further, the developed crack growth rate expression also explains the effect of specimen size and loading frequency on the fatigue life of structural members. 2. Model formulation Previous studies, (Le and Bažant (2011), Bažant, Le and Bazant (2009)) have shown that the macro-scale fracture originates with the breakage of interatomic bonds at the nano-scale and subsequently creates micro-cracks. Therefore, the structural failure can be correlated with the statics of interatomic bond breakage (Zhaodong and Jie (2018)). In this study, a theoretical approach developed by Le and Bažant, (2011) of atomic bond breakage due to random jumps of thermally energetic atoms over the activation energy barrier has been followed.

Fig. 1. Hierarchical structure of concrete fracture.

1.1. Jump frequency Consider a traction free crack length of ‘ a ’ in concrete is surrounded by many micro cracks ( a m i , i = 1, 2, 3..) at the tip along the direction of crack path in its FPZ while loading. Each micro crack contains its own FPZ of nano elements proposed by Le and Bažant, (2011) . Assuming a nano element of length, a n positioned at the tip of the micro crack having average length, a m in the direction of crack propagation as shown in Fig. 1. These nanoscale elements can be considered an atomic lattice block of completely disordered nanoparticles of the calcium silicate hydrate in concrete. The C-S-H system offers an adhesion force that creates an energy barrier for nanoparticles, which may be considered an activation energy barrier to interatomic bonds' breakage at the micro crack tip (Zhaodong and Jie (2018)). This energy barrier can act as an obstruction in the form of required energy release of concrete when the micro crack tip moves. Since the thermally activated nanoparticles are used to be in an energy equilibrium state, they might drive C-S-H’s disordered nanoparticles system to cross the activation energy barrier after getting energy from the external source. Under the action of external loading, it disturbs the energy equilibrium between two adjacent metastable states, resulting in the breakage of interatomic bonds between nanoparticles and promoting micro crack movement in the direction of its crack path (Zhaodong and Jie (2018)). The energy consumed ‘∆Q’ , in the crack front advancement ‘δ’ can be written as following .