PSI - Issue 13

Leonardo Giangiulio Ferreira de Andrade et al. / Procedia Structural Integrity 13 (2018) 1908–1914 Leonardo G. F. Andrade and Gustavo H. B. Donato / Structural Integrity Procedia 00 (2018) 000–000

1911

4

and elastic-plastic (to assess plasticity effects) analyses employed the same meshes for each geometry. To simulate varying crack curvatures, a specific algorithm was developed to adapt crack tip meshes to varying tunneling levels, denoted here T ; as illustrated in Fig. 1(d-f) , T is defined, for each specimen, as T = max(a/W) - min(a/W) . Results are presented in terms of T and also T/B . Loading was applied to the models as incremental displacements (Load Line Displacements - LLD ) at the central back nodes of the crack plane representing the center roller of the three-point bending device. For the tunneling study, an elastic loading and unloading of 0.2mm was configured. For the plasticity study, in its turn, a special loading scheme was created including several loading and unloadings as follows: i) the specimen was initially loaded until LLD = 0.2mm employing 100 steps; ii) in the sequence, the specimen was partially unloaded by -0.1mm LLD employing 100 steps; iii) the specimen was, then, reloaded adding 0.3 mm LLD using the same step size of previous loading; iv) this procedure was repeated until LLD = 5mm was reached, which represented an average of 24 partial unloading steps in each sample. After all models were processed and its results (in terms of load P vs. CMOD - V ) evaluated, elastic unloading compliance ( C = V/P ) could be quantified for each studied condition (tunneling or plasticity levels). Every result was compared to the respective T = 0 or absence of plasticity conditions (see Fig. 1(d) ), respectively clarifying the effects of tunneling and plasticity on elastic unloading compliance and, consequently, on its instantaneous crack size predictions. The quantitative effect on compliance is denoted C DELTA ( Eq. 3 ), where C is the compliance and C 0 is the reference compliance (considering no tunneling or no plasticity as explained earlier). C DELTA = � C - C 0 � C 0 .100 (3)

0 100 200 300 400 500 600 700 800 900

T

n5 n10 n20

Crack front

Stress (MPa)

Notch

y

T = 0 Straigth crack

T ≠ 0 Tunneled cracks

(d)

(b) c)

x

z

0

0.1 0.2 0.3 0.4 Strain (mm/mm)

(a) b

(a)

Fig. 1. (a) Stress-strain curves for considered materials; (b) example of a SE(B) model; (c) blunt crack tip with a spider web mesh detail and; (d) examples of straight and tunneled cracks, with varying T levels. 3. Results 3.1. Crack tunneling effects on elastic unloading compliance C Figures 2(a-b) display C DELTA as a function of the tunneling level T and relative tunneling level normalized by the thickness T/B for B=25.4mm ( W/B=2 ) and B=12.7mm ( W/B=4 ) respectively. The ASTM E1820 (2018) limit for maximum tunneling is displayed as a vertical dashed line in the graphics. Details can be found on the aforementioned standard and won´t be presented here due to space limitations. It can be observed in Figs. 2(a-b) that within the ASTM limits compliance variation C DELTA is within ±0.5% . The only case in which C DELTA presented a higher deviation was B=50.8mm and W/B=1 with a/W=0.2 (not presented here), where C DELTA was -1.63% . When those limits are violated, C DELTA presents a strong decrease reaching -13.67% for deep crack in W/B = 4 . The decrease leads to a predicted crack smaller than ASTM equivalent. In all three thicknesses, a/W=0.5 (medium crack) had the less intense effect of tunneling, but, in general, the obtained results prove that crack tunneling effects are remarkable. Evaluating now Figs. 2(c-d) , the open markers reveal the deviation between compliance predicted by ASTM based on the crack size computed by Eq. (5) to the numerical compliance obtained directly from the refined 3D model containing the tunneled crack. In this equation, and assuming that the crack front curvature is symmetric, β coefficients