PSI- Issue 9

Riccardo Fincato et al. / Procedia Structural Integrity 9 (2018) 126–135 Author name / Structural Integrity Procedia 00 (2018) 000 – 000 Autho name / Structural Integrity Procedia 00 (2018) 000 – 000

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Figure 1d, showing a good agreement with the test conducted in Li et al. (2016). The following Table 1 reports the elastoplastic parameters for the Damage S-S model, where u , c , χ and R e are four constants proper of the subloading surface theory. u regulates the smooth elastoplastic transition between the sub-yield and fully plastic stress state. c and χ regulate the speed of the similarity centre in the stress space and R e introduces a small elastic domain (Tsutsumi et al., 2006). The six damage parameters for the Lemaitre’s appro ach are reported in Table 2, whereas the five material parameters in Eqs. (8)-(9) are reported in Table 3. It is worth mentioning that the calibration of the damage parameters cannot be done entirely with a single uniaxial tensile test since it would define only one point on the failure envelope. The stress triaxiality and the Lode angle dependency have to be defined by the calibration of at least two additional tests, one on a notch round bar and one on a flat grooved plates. The black line in Figure 3d refers to the damage data already calibrated. The calibration is discussed in the following section 3.3. Figure 1d, showing a good agreement with the test conducted in Li et al. (2016). The following Table 1 reports the elastoplastic parameters for the Damage S-S model, where u , c , χ and R e are four constants proper of the subloading surface theory. u regulates the smooth elastoplastic transition between the sub-yield and fully plastic stress state. c and χ regulate the speed of the similarity centre in the stress space and R e introduces a small elastic domain (Tsutsumi et al., 2006). The six damage parameters for the Lemaitre’s appro ach are reported in Table 2, whereas the five material parameters in Eqs. (8)-(9) are reported in Table 3. It is worth mentioning that the calibration of the damage parameters cannot be done entirely with a single uniaxial tensile test since it would define only one point on the failure envelope. The stress triaxiality and the Lode angle dependency have to be defined by the calibration of at least two additional tests, one on a notch round bar and one on a flat grooved plates. The black line in Figure 3d refers to the damage data already calibrated. The calibration is discussed in the following section 3.3.

Table 1. Elastoplastic parameters for the Damage S-S. Young’s modulus Table 1. Elastoplastic parameters for the Damage S-S. Young’s modulus

222700 [MPa] 222700 [MPa] 381.884 [MPa] 381.884 [MPa] 969.14 [MPa], 0.2 969.14 [MPa], 0.2 0.3 0.3 1000 1000

Poisson’s ratio Poisson’s ratio

u u F 0 R e F 0 R e

0.2 0.2 200 0.9 200 0.9 1.5 1.5 0.01 0.55 0.01 0.55 1.0 1.0 0.85 0.85

K , n K , n

c χ c χ

Table 2 . Damage parameters for the modified Lemaitre’s ductile damage law s 1 Table 2 . Damage parameters for the modified Lemaitre’s ductile damage law s 1

9.5 [MPa] 9.5 [MPa]

s 2 s 3 s 2 s 3

β (Fincato and Tsutsumi, 2017a) α 1 (Fincato and Tsutsumi, 2017a) α 2 (Fincato and Tsutsumi, 2017a) β (Fincato and Tsutsu i, 2017a) α 1 (Fincato and sutsu i, 2017a) α 2 (Fincato and Tsutsumi, 2017a)

Table 3. Damage parameters for the modified Mohr- Coulomb’s law. A Table 3. Damage parameters for the modified Mohr- Coulomb’s law. A

279.5 [Mpa] 279.5 [Mpa] 0.285 0.285 360 [Mpa] 360 [Mpa] 0.3 0.3 0.1 0.1

N c 1 c 2 d 1 N c 1 c 2 d 1

3.2. Tensile tests 3.2. Tensile tests

The load-displacement curves obtained in the numerical simulation are reported in Figure 4. All the analyses were conducted imposing a prescribed displacement boundary condition pulling the samples until the damage reached a critical value D c . The limitation to the damage parameter was included in order to avoid the pathological mesh dependency due to the damage localization. In the previous work Fincato and Tsutsumi (2017a) set the critical damage to 0.3, and in the present work a critical damage D c = 0.2 is chosen. The graphs in Figure 4a, b and c report the analyses carried out on the notched round bars, whereas the ones in Figure 4d, e and f refer to the flat grooved plates. The experimental solutions are reported with squared blue marks and the blue crosses indicate the point where the cracks were detected on the samples. In addition to the green (Lemaitre’s ductile damage law) and purple (Mohr - Coulomb’s failure criterion) lines, the black curves report the load-displacement solutions obtained without ductile damage. As it can be seen the solutions without damage can catch quite realistically the load peaks in the notched bar cases, however, the maximum loads seem to take place at higher displacements in case of the flat grooved plates. Moreover, the simulation without the ductile damage cannot catch the post necking behaviour, overestimating the material performances. The same tendency is detected in the numerical solution of the flat plates in Li et al. (2016), reported in Figure 4d e and f with black dashed lines. The load-displacement curves obtained in the numerical simulation are reported in Figure 4. All the analyses were conducted imposing a prescribed displacement boundary condition pulling the samples until the damage reached a critical value D c . The limitation to the damage parameter was included in order to avoid the pathological mesh dependency due to the damage localization. In the previous work Fincato and Tsutsumi (2017a) set the critical damage to 0.3, and in the present work a critical damage D c = 0.2 is chosen. The graphs in Figure 4a, b and c report the analyses carried out on the notched round bars, whereas the ones in Figure 4d, e and f refer to the flat grooved plates. The experimental solutions are reported with squared blue marks and the blue crosses indicate the point where the cracks were detected on the samples. In addition to the green (Lemaitre’s ductile damage law) and purple (Mohr - Coulo b’s failure criterion) lines, the black curves report the load-displacement solutions obtained without ductile damage. As it can be seen the solutions without damage can catch quite realistically the load peaks in the notched bar cases, however, the maxi um loads seem to take place at higher displacements in case of the flat grooved plates. Moreover, the simulation without the ductile damage cannot catch the post necking behaviour, overestimating the material performances. The same tendency is detected in the numerical solution of the flat plates in Li et al. (2016), reported in Figure 4d e and f with black dashed lines.