PSI - Issue 35

S. YaŞayanlar et al. / Procedia Structural Integrity 35 (2022) 18– 24 Yas¸ayanlar et al. / Structural Integrity Procedia 00 (2021) 000–000

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same location, the way it evolves spatially is quite di ff erent. It supports similar observations reported in the literature Poh and Sun (2017) regarding the spatial evolution of the localization band. 4.2. V-notched Bar The V-notched specimen of Li et al. (2011) which was loaded in tension uni-axially is analyzed using the proposed tetrahedra element and the localizing implicit gradient damage formulation. In this problem, von Mises plasticity is used which was also investigated by Mororo and van der Meer (2020) and Miehe et al. (2016) using 2D models. The geometry of the specimen along with the boundary conditions are shown in Figure 3 and the model parameters are listed in Table 2. Using symmetry boundary conditions, the upper half of the specimen is discretized with three-field tetrahedra elements. First a mesh convergence study is conducted and the resulting force-displacement graphs are shown in Figure 4. These results once again verify the convergence of implicit gradient type non-local formulations upon mesh refinement. In Figure 4, load-displacement graph of the fine mesh is compared with the experimental results of Li et al. (2011) and the predictions of Miehe et al. (2016) which uses a phase-field formulation.

Table 2: Model parameters of V-notched specimen

Model Parameter

Value

l2

E

68900 MPa

0.33

ν

eqv pl )

r

0 . 12 MPa

Hardening law

850 × (0 . 3 + 1.5 mm

u

l3

l4

l c κ i κ c

0.025

0.25 0.02 2.35 0.25

l1

α β R

Fig. 3: Geometry and the boundary conditions of V-notched specimen

1.0

η

The agreement between the experiments and the predictions of the current implementation is good except the tail of the curve where the evolution of damage slows down significantly. The same analysis is repeated using the two field element formulation (quadratic displacement and linear non-local equivalent plastic strain) in which the pressure interpolation is omitted and calculated from the volumetric strain ( e ) as obtained from the resulting displacement field. The analysis diverged at a maximum damage level of approximately 0.75 and the resulting force-displacement curve is given in Figure 4.

4000

4000

Li et al. (Experimental) Miehe et. al (Phase field) Current Formulation (LIGD) Two−field Formulation

Fine Medium Coarse

3500

3500

3000

3000

2500

2500

2000

2000

1500 Force (N)

1500 Force (N)

1000

1000

500

500

0

0

0

0.1

0.2

0.3

0.4

0.5

0

0.1

0.2

0.3

0.4

0.5

Displacement (mm)

Displacement (mm)

Fig. 4: Left: Mesh convergence study; Right: Comparison of LIGD predictions, phase-field predictions and experimental results

Although the curve is very close to the curve obtained by three-field formulation, the pressure distributions as shown in Figure 5 are completely di ff erent. The two-field formulation results in a non-smooth pressure distribution with very large compressive and tensile values (as compared to three-field formulation) whereas the corresponding distribution is smooth in case of three-field formulation.