PSI - Issue 39

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

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2 (1) Here, G m , K m , and E m are the energy release rate, stress intensity factor (SIF), and modulus of elasticity at the microscale, respectively. g( γ) is the microscale geometric factor depends on average micro crack length and its critical size. According to the transition rate process theory, ( Le and Bažant, (2011)) the net transfer frequency between two metastable states in the forward direction can be expressed using Kramer’s formula (Le and Bažant (2011)) . ( ) 0 0 ( )/ ( )/ 2 1 / Q Q kT Q Q kT T m m f e e K E ν ν − −∆ − +∆ = − = (2) Here, ν T =kT/h , is characteristic attempt frequency rate for reversible transition, ν 1 = 2ν T e � - QkT0 � δg(γ) kT is a material constant, h is Plank constant, T is absolute temperature and, k is Boltzmann constant. 1.2. Micro crack growth rate Linear elastic fracture mechanics (LEFM) can be applied to the scattered individual micro crack considering brittle nature to quantify its extension rate with the net forward jump, ‘ f δ ’ of nanoparticles after applying external load. The partially differentiated crack opening equation with respect to time ‘ t ’ can be written as following. 2 2 4 2 ( ) ( ) ( ) ( ) m m m m m a a COD t E t a x σ π ∂ ∂ = ∂ − ∂ (3) Here, σ m is the stress, a m is the average micro crack length, and E m is the modulus of elasticity at microscale. For a small value of COD, and a m , ( ) and ( ) can be written as, ( ) and ( ) , respectively. Here, ( ) will be the crack opening ‘ f δ ’. The C-S-H gel arrangement contains water molecules, calcium, and silica tetrahedron in cement hydrates. The initial undisturbed arrangement of these primary particles at the micro crack tip used to be well arranged and form a local free energy state, (Zhaodong and Jie (2018)). However, in the case of externally applied fatigue loading, the repeating cyclic action would disturb the relative regular system of C-S-H and make a permanent microstructural change at the local crack tip. Therefore, a permanent micro crack advance can be expressed by replacing K m with the factored stress intensity factor Δ K m in fatigue loading condition having cycle time t c . Further the micro crack growth rate with respect to loading cycle can be accounted by integrating the crack growth rate over a t c cyclic loading period. For the new extended micro crack length ‘ a m +a n ’ the above equation can be simplified as following by substituting, ν 3 = √ �π 3 a n 32 � ν 1 δ t c � R σ 3 � , where, R σ is the stress ratio. Study (Simon and Chandra Kishen, (2017)) suggest that the crack induced between cement paste and aggregate grow to a certain critical micro crack length, l m during loading, and further, try to coalesce with the existing macro crack resulting traction free crack increment. Accordingly, SIF gets modified at the macro-crack tip. A similar concept can be used to express the factored stress intensity factor, Δ K m at micro crack tip in terms of factored stress intensity factor, αΔ K at macroscale, where, α depends on geometry and material property of both the scales. ( ) m m m Q G K E g δ δ γ ∆ = = / ( ) 0 ( ) ( ) m c t m da dt  d a d N dt   =    ∫ = 3 / m m K E ν ∆ (4)