PSI - Issue 23

Tomáš Oplt et al. / Procedia Structural Integrity 23 (2019) 101–106 Tomáš Oplt / Structural Integrity Procedia 00 ( 2019) 000 – 000

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Plasticity induced crack closure is happening due to the residual plastic deformations, which forms the plastic wake behind the growing fatigue crack (Elber 1970; Suresh 1998). In order to simplify the calculation, but provide reliable results at the same time, it is recommended to advance the crack through at least one initial plastic zone size (Singh et al. 2008; Solanki et al. 2004b). Incrementing the fatigue crack was simulated by releasing the crack front nodes at the moment of maximum load. One crack length increment ∆ a equals to the size of element at the crack tip L e , therefore advancing through the size of one plastic zone leads to 10 cycles. One cycle consists of three parts – loading, debonding and unloading (LDU). In order to remove anomalous contact of first node behind the crack tip (Singh et al. 2008), in the end crack growth process one aditional load and unload without debonding was added (LDULU). For the purpose of the crack faces premature contact simulation, contact elements were employed in the analysis with penalty method algorithm with penalty factor 10 and penetration tolerance factor 0.1. Closure levels were determined after the last cycle of simulation, when sufficiently long plastic wake ensures that closure values were stabilized. Most common determination method used in 2D analyses is the monitoring of the first node behind the crack tip (Baptista et al. 2017; McClung and Sehitoglu 1989; Oplt et al. 2019; Solanki et al. 2004a). Changes of the 1 st node displacement value u y from positive to negative indicates penetration into the opposite crack face, thus that crack was closed (see Fig. 2b). Displacement equals to zero at the substep n *, which is interpolated value between substeps n and n +1. Then, effective stress intensity factor range ΔK eff , during which the crack is fully opened and thus, is promoted to grow, can be calculated according to eq. 2. In order to respect the SIF distribution affected by the free surface, it is necessary to perform linearly elastic calculation to obtain the distribution for maximum SIF K max . For that purpose, J-integral was defined along the crack front of the specimen with crack length a f = 15 mm and K max was estimated at each node of the crack front. Afterwards, value of the closure SIF K cl , at which the crack was closed, was determined (eq. 3). Same approach was performed for all nodes along the crack front for both LDU and LDULU scheme. In contrary to the e.g. stiffness method, where the stiffness change during unloading is assessed and provides global information about the closure, first node displacement method provides information about closure locally at each point along the crack front. Δ = ∗ ⋅ Δ (2) = , − Δ (3) 3. Results and discussion Fig. 3 presents the distributions of evaluated parameters through thickness for case of load ratio R = 0. Free surface has a strong impact on elastic SIF K max as it decreases at the vicinity of the free surface while in the rest of the body remains constant. At the vicinity of the free surface, SIF and FCPR are lower then in the middle of the body and therefore, growing crack will be curved from the initial straight crack until it reaches characteristic crack curvature. Detailed study of the free surface effect was published by (Oplt et al. 2018). Second major influence on the fatigue crack front curvature causes the presence of the crack closure phenomena. In general, crack closure partly retards the growing crack. However, retardation is not constant through the whole thickness but differs again at the free surface area. When the body was unloaded from the maximum load K max = 21.2 MPa√m , premature closure appeared in the middle of the body at the value K cl (LDU) = 6.6 MPa√m determined from LDU scheme and at K cl (LDULU) = 4.5 MPa√m from LDULU scheme. At free surface was the range even shorter, unloading started from K max = 17 MPa√m and closure appeared at K cl (LDU) = 9.9 MPa√m or K cl (LDULU) = 8.8 MPa√m , respectively. In summary, crack closure causes retardation of the crack growth. Instead of advancing the crack by the full load range ∆ K = 21.2 MPa√m , for which the crack is supposed to be open, crack is effectively loaded only by the range ∆ K eff (LDU) = 14.6 MPa√m or ∆ K eff (LDULU) = 16.7 MPa√m at the most of the body due premature closure. Closure occurs even earlier at the free surface, where the effective value only ∆ K eff (LDU) = 7.1 MPa√m or ∆ K eff (LDULU) = 8.2 MPa√m . For the purpose of comparison, horizontal line of theoretical ∆ K eff = 15 MPa√m for the load range R = 0 according to Newman’s empirical equation (Oplt et al. 2019) was added.