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

1910

3

average of the points closer to the edges followed by an average of the eight remaining values is done to determine the equivalent crack size. More details can be found in the aforementioned standard ( ASTM E1820 , 2018). Studies regarding plasticity effects on elastic unloading compliance are scarce. Tobler and Carpenter (1985) were among the first that published a study considering this theme. Compact under Tension - C(T) - specimens were modeled using 2D techniques. Compliance was then estimated, and the greater the loading, greater the compliance deviation would be, with deviations reaching up to 5% . Additionally, Xu, Shen and Tyson (2005) studied this effect on SE(B) specimens. This study was based on a previous research conducted by Jiang et al. (2004) for the same geometry. Refined 3D models for SE(B) specimens were developed and results were compared with real test data available from Jiang et al. (2004), with results showing divergences of 9% (high resistance steels) and 12% (aluminum alloys). Recently, plasticity effects were studied by Vestraete (2014) for specimens under tension - results showed that compliance would be reduced in the beginning of the loading, following by an increase. In this context, this research developed very refined models and a systematic investigation regarding effects of tunneling and plasticity on the EUC technique applied to SE(B) specimens of varying geometrical features. 2. Materials and numerical procedures 2.1. Material constitutive models To evaluate only crack tunneling effects on compliance, no plasticity is desirable on the specimen to isolate the phenomenon. Based on that, a material following Hooke’s Law was judged enough to represent structural steels in the linear elastic regime - E=206 GPa and ν =0.3 were considered. On the other hand, the assessment of plasticity effects demand elastic-plastic material models. Thus, analyses employed an elastic-plastic constitutive model with J 2 flow theory and conventional Mises plasticity in Large Geometry Change ( LGC ) setting. The numerical solutions employ a simple power-hardening model to characterize the uniaxial true stress-logarithmic strain ( Eqs.1-2 ). The material properties considered are typical of structural steels and are presented by Tab.1 ; here, σ ys and ε ys are the yield stress and strain ( 0.2% offset), and n is the strain hardening exponent. The true stress-strain curves for the considered materials can be seen in Fig. 1(a) .

0 ఌ ఌ ೤ೞ ൌ ఙ ఙ ೤ೞ , ߝ ൏ ߝ ௬௦ ఌ ఌ ೤ೞ ൌ ఙ ఙ ೤ೞ ௡ , ߝ ൒ ߝ ௬௦ 0 0 ,         0   0 0 ,             n

Table 1. Materials considered for the study. Material Name E (GPa) ν σ ys (MPa)

n

(1)

n5

206

0.3

257

5

n10 n20

206 206

0.3 0.3

412 687

10 20

(2)

2.2. Numerical procedures and FEM models The developed 3D finite element models were based on previous refined modeling conducted and validated in the research group by Moreira and Donato (2010, 2013). MSC Patran 2013 was the pre-processor of choice and Abaqus CAE 6.13 was used as processor and post-processor. The specimen being investigated is the Single-Edge notched under Bending - SE(B) . Three specimen proportions were simulated with varying width to thickness ratios W/B . Width was fixed equal to 50.8mm , while three thickness were modeled ( 50.8 , 25.4 and 12.7mm ), leading to W/B ratios of 1 , 2 and 4 . The specimen with W/B=2 represents the 1T dimensions in accordance with ASTM E1820 (2018). Also, three values of relative crack depths ( a/W = 0.2 , 0.5 and 0.7 ) were modeled to represent shallow, medium and deep cracks. No crack growth was implemented in the aforementioned models, and only one quarter of the specimens were modeled with appropriate boundary conditions to ensure symmetry, saving computational resources. An example of an SE(B) model is displayed in Fig. 2(b) . In addition, work done by other researchers corroborate with this design (Huang and Zhou (2015), Yan and Zhou (2014)). Meshes were designed including a blunt crack tip with element sizes and pattern following a spider web mesh ( Fig. 2(c) ). While Moreira employed a crack tip radius of 0.0005 mm for elastic compliance analyses, such radius did not prove to be appropriate for plasticity evaluations. Consequently, based on a convergence analysis and supported by the literature (Verstraete, 2014), the authors detected that a blunt mesh with 0.05 mm of crack tip radius could be employed with negligible deviation and, at the same time, ensuring numerical convergence for high loads (and plasticity) scenarios. For consistency purposes, all elastic (to asses tunneling effects)