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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3. Upscaling Under fatigue loading condition in concrete, FPZ is primarily governed by micro-cracks, and can be considered as the main reason of its toughening mechanism. Therefore, according to energy balance principle, energy consumed by a macro crack for unit crack growth will be equal to the energy consumed by all the active lower scale cracks present in FPZ in a unit crack growth (Le, Manning and Labuz (2014)). (5) Here, G m i and d(a m i d(N) ) are energy release rate and crack growth rate of an ith micro crack in concrete’s FPZ having n m active icro cracks. G If c and d d ( ( N a) ) are the critical energy release rate and crack growth rate for infinite specimen size at macro-scale. Having shielding and amplifying of micro-cracks inside FPZ (Kachanov (1993),Wang (2019)), we can consider the average value of energy release rate and crack growth rate at micro-scale, G m and d(a m d(N) ) respectively. Inside the FPZ, not every micro-crack participate in the formation of final crack path, many undergoes unloading. The micro-cracks, those actually take part in coalescence and finally goes under traction free crack formation, called active number of micro cracks (Le, Manning and Labuz (2014)), and can be expressed as, n m = � ΔK 2 EG If c � m . Here, m is the power exponent which depends on material as well as geometric property, and E is the modulus of elasticity at macro-scale. Considering, f c = t 1 c is the loading frequency, the crack growth rate expression for the infinite specimen size can be written as following. ( ) 3/2 2 2 1 3 2 2 ( ) 1 8 ( ) ( ) m eff m m If m c a d a K E g d N l K E f ν β +     ∆ =         (6) Here, K If is the fracture toughness for infinite specimen size, which can be replaced as K If � D+ D D 0 � 0.5 for any specimen size D , where D 0 is the transition size, which can also be considered as aggregate size (Bazant and Kazemi (1990)). l m is the critical length of micro crack, which will depend on specimen size. a eff is the effective crack length, 3 / 2 ( ) , eff eff m eff a a l g g D a β   − =       , (Simon and Chandra Kishen (2017)). Therefore, the final crack growth rate expression for any specimen size can be written as following. ( ) 1 3 3 0 2 1 2 ( ) 1 ( ( )) n n n eff n c D D da C a g dN D f σ β − + + + = ∆ (7) Here, C 1 is purely a material property, Δσ is the factored stress amplitude, and n=2m+1 . The above crack growth rate expression is the function of material property, specimen size, size of aggregate used, and the externally applied factored stress amplitude with its loading frequency. 4. Calibration, Validation and Results The developed expression for crack growth rate Eqn. (6) has been calibrated with the experimental results of Bhowmik and Ray, (2019) of flexural cyclic loading condition on three different beam size specimen 50mm, 100mm 1 = = ∑ ( ) m ( ) ( ) ( ) i m i c n If m i d a G d a d N G d N