Calibration of cohesive parameters for fracture test simulations of sub-sized bend specimens of AM steels

Abstract

Crack extensions in sub-sized side-grooved bend specimens of AM steels are examined with changing cohesive parameters. Two-dimensional FE analyses are first conducted to simulate crack extensions by nodal release. The simulation results indicate the near-tip maximum opening stress and separation work rate increase and then slightly decrease for the bend specimens. The hydrogen-charged bend specimens have lower near-tip maximum opening stresses and average separation work rates than those of the uncharged bend specimens. The changing near-tip maximum opening stresses and the separation work rates with increasing crack extension are determined as references for the changing cohesive strengths and energies, respectively. Two-dimensional FE analyses with different changing cohesive parameters are then performed. The simulation results with calibrated cohesive parameters can match well with the test results. The hydrogen-charged specimens have the lower changing cohesive strength vs crack growth curves compared with those of the uncharged specimens. The hydrogen-charged specimens have either lower or similar changing cohesive energy vs crack growth curves compared with those of the uncharged specimens. Finally, a three-dimensional FE analysis is conducted to simulate crack extension by nodal release in one bend specimen to examine the stress triaxialities in front of the growing crack fronts at different locations from the center plane to the planes near the side groove front with increasing crack extension to identify possible physical reasons for the changing cohesive parameters.

Appendix A: 2-D FE analyses to simulate crack extensions by nodal release

Figure 

Fig. 23

The FE model of the right half of the LS-2 specimen

23 shows the FE model of the right half of the LS-2 specimen. The FE model shown in Fig. 23 is the same as the one shown in Fig. 6 except only the right half of the specimen without cohesive elements is modeled due to the symmetry conditions. Figure 23 shows the symmetry condition, marked in blue, applied to the ligament. The FE models for the other three bend specimens are similar and will not be shown here. The crack tip opening displacement (CTOD) is in the order of $J_{0.2}$/$\sigma_{0}$. The size of the elements of 0.1 mm by 0.1 mm near the crack plane is in the order of the CTOD in order to determine the near-tip maximum opening stress at about one element away from the growing crack tip with the geometry nonlinearity.

The boundary conditions of the FE analyses are controlled by the multiple-linear $\Delta a$ vs pin displacement curves. The crack extension is controlled by decreasing the length of the symmetry condition. The available experimental displacement data are the CMODs. The pin displacement boundary is applied to the center of the loading pin. Several rounds of the iterations are necessary to obtain the crack extension vs pin displacement curves to give the multiple-linear $\Delta a$ vs CMOD curves measured during the experiments. Figures 

Fig. 24

The experimental and computational $\Delta a$ vs CMOD curves for the a LS-2 specimen and b LS-4 specimen

24a and b show the computational multiple-linear $\Delta a$ vs CMOD curves for the LS-2 and LS-4 specimens, respectively. The computational multiple-linear $\Delta a$ vs CMOD curve for the 3-D FE analysis for the LS-2 specimen as presented in Sect. 5 is also shown in Fig. 24a. It should be mentioned that the CMODs from the 3-D FE analysis are relatively uniform through the thickness of the specimen along the Z direction. Therefore, the CMODs are taken from the specimen surface from the 3-D FE analysis. Figures 

Fig. 25

The experimental and computational $\Delta a$ vs CMOD curves for the a TS-2 specimen and b TS-4 specimen

25a and b show the computational multiple-linear $\Delta a$ vs CMOD curves for the TS-2 and TS-4 specimens, respectively. Figures 24 and 25 show that the entire simulation is divided into several steps. During each step, the loading pin is moving downward and a finite length of the symmetry condition (the remaining ligament) is also released to propagate the crack to give the computational $\Delta a$ vs CMOD curve for comparison with the test results.

The changing cohesive strengths can be referenced by examining the changing near-tip maximum opening stresses. The opening stresses $\sigma_{xx}$’s in front of the crack tips at the ends of crack extension steps for the LS-2 specimen is shown in Fig. 

Fig. 26

The opening stresses and near-tip maximum opening stresses for the LS-2 specimen

26. Here, the black dotted lines represent the opening stresses in front of the crack tips and the red solid line represents the connections of the near-tip maximum opening stress locations in front of the growing crack tip. The $\sigma_{xx}$ distributions for the LS-4, TS-2 and TS-4 bend specimens are similar and will not be shown here. However, for convenience of comparison, Fig. 

Fig. 27

a The near-tip maximum opening stresses and their locations and b the separation work rates for the LS-2 specimen and LS-4 specimen

27a shows the near-tip maximum opening stresses and their locations at the ends of the crack extension steps for the LS-2 and LS-4 specimens. Here, the near-tip maximum opening stresses increase from the small values below 1,000 MPa near crack initiation to the large values about 1,815 MPa and 1,587 MPa at about 1 mm in front of the initial crack tips and then decrease with increasing crack extension for the LS-2 and LS-4 specimens, respectively. The near-tip maximum opening stresses for the LS-2 specimen in Fig. 27a from the 2-D FE analysis are similar to those from the 3-D FE analysis shown in Fig. 19. For comparison, the near-tip maximum opening stresses along Path 1 (center plane) for the LS-2 specimen from the 3-D FE analysis are shown in Fig. 27a. As shown in Fig. 27a, the near-tip maximum opening stresses of the LS-4 specimen is lower than those of the LS-2 specimen.

The changing cohesive energies can be referenced by examining the changing separation work rates in the 2-D plane strain FE analyses of the half specimens as explained in the following. Here, the balance of the energies of the system can be written as

$$ W_{{{\text{ext}}}} = U_{e} + W_{p} + W_{s} $$

(A1)

where $W_{{{\text{ext}}}}$ is the external work, $U_{e}$ is the elastic strain energy, $W_{p}$ is the plastic dissipation, and $W_{s}$ is the separation work for creating a new free surface. For each step, Equation (A1) can be written as

$$ \Delta W_{{{\text{ext}}}} = \Delta U_{e} + \Delta W_{p} + \Delta W_{s} $$

(A2)

where $\Delta W_{{{\text{ext}}}}$ is the change of external work, $\Delta U_{e}$ is the change of elastic strain energy, $\Delta W_{p}$ is the change of plastic dissipation, and $\Delta W_{s}$ is the change of separation work for creating a new free surface. For each step with a crack area increment $\Delta A$, Equation (A2) can be written as

$$ \frac{{\Delta W_{{{\text{ext}}}} }}{\Delta A} = \frac{{\Delta U_{e} }}{\Delta A} + \frac{{\Delta W_{p} }}{\Delta A} + \frac{{\Delta W_{s} }}{\Delta A} $$

(A3)

Here, $\frac{{\Delta W_{{{\text{ext}}}} }}{\Delta A}$ represents the average external work rate for a crack area increment, $\frac{{\Delta U_{e} }}{\Delta A}$ represents the average elastic strain energy rate for a crack area increment, $\frac{{\Delta W_{p} }}{\Delta A}$ represents the average plastic dissipation rate for a crack area increment, and $\frac{{\Delta W_{s} }}{\Delta A}$ represents the average separation work rate for a crack area increment.

It should be mentioned that in ABAQUS, the change of separation work, $\Delta W_{s}$, can be calculated by using the nodal forces and nodal displacements at the released nodes for each step. The calculated separation work rate can also be validated by Equation (A3) from the values of $\frac{{\Delta W_{{{\text{ext}}}} }}{\Delta A}$, $\frac{{\Delta U_{e} }}{\Delta A}$, and $\frac{{\Delta W_{p} }}{\Delta A}$ for the known change of crack area in each step. The separation work rates per unit increase of crack area are plotted for the crack extension steps for the LS-2 and LS-4 specimens in Fig. 27b. As shown in Fig. 27b, the separation work rates increase from small values to reach the maximum values near the crack extensions of about 1.0 mm and 0.6 mm for the LS-2 and LS-4 specimens, respectively. The separation work rates then decrease with increasing crack extension. In Fig. 27b, the separation work rates for the LS-2 specimen from the 3-D FE analysis of the uniform crack extension by nodal release are also shown. The 3-D FE analysis is discussed in Sect. 5. As shown in Fig. 27b, the trends of the separation work rates from both 2-D and 3-D FE analyses are very similar.

The average separation work rate $\overline{{\dot{W}_{s} }}$ is defined as

$$ \overline{{\dot{W}_{s} }} = \frac{{\sum {\Delta W_{s} } }}{{\left( {\Delta a} \right)_{f} B_{n} }} $$

(A4)

where $\sum {\Delta W_{s} }$ represents the summation of the separation works for all steps, $\left( {\Delta a} \right)_{f}$ represents the final crack growth of the simulation, and $B_{n}$ represents the net thickness of the specimen. The values of the average separation work rates of the LS-2 and LS-4 specimens are also plotted in Fig. 27b. The average separation work rate of the LS-4 specimen is lower than that of the LS-2 specimen.

Similarly, Fig. 

Fig. 28

a The near-tip maximum opening stresses and their locations and b the separation work rates for the TS-2 specimen nd TS-4 specimen

28a and b show the near-tip maximum opening stresses and separation work rates for the TS-2 and TS-4 specimens, respectively. Figure 28a shows that the near-tip maximum opening stresses increase from the small values about 1,000 MPa near cack initiation to the large values about 1,764 MPa and 1,624 MPa near the crack extension of about 1.3 mm in front of the initial crack tips and then decrease with increasing crack extension for the TS-2 and TS-4 specimens, respectively. Similarly, the near-tip maximum opening stresses of the TS-4 specimen is lower than those of the TS-2 specimen. Figure 28b shows that the separation work rates increase from small values to reach the maximum values near the crack extensions of about 0.6 mm and 0.8 mm for the TS-2 and TS-4 specimens, respectively. The separation work rates then decrease with increasing crack extension. Similarly, the values of the average separation work rates of the TS-2 and TS-4 specimens are also plotted in Fig. 28b. The average separation work rate of the TS-4 specimen is lower than that of the TS-2 specimen.

It should be mentioned that in the CZM approach, the changing cohesive strengths of the cohesive elements should be lower than or close to the near-tip maximum opening stresses. Otherwise, the cohesive elements ahead of the growing crack tips will not open up as suggested in Sung et al. (2021a). Since the separation of the cohesive elements ahead of the growing crack tips results in less plastic dissipation in the neighboring continuum elements, the separation work rate obtained from the 2-D FE analysis is not the same as the cohesive energy. However, the trend of increasing and decreasing separation work rate with increasing crack extension can still be used as reference for the calibration of the changing cohesive energy.

The lower near-tip maximum opening stresses and average separation work rates of the hydrogen-charged specimens when compared with those of the uncharged specimens indicate the effect of hydrogen on the AM steels. Also, a comparison of the near-tip maximum opening stresses in Figs. 27a and 28a indicate that the uncharged LS specimen has higher near-tip maximum opening stresses than those of the uncharged TS specimen. The hydrogen-charged LS specimen has slightly lower near-tip maximum opening stresses than those of the hydrogen-charged TS specimen. Also, Figs. 27b and 28b indicate that the uncharged LS specimen has lower separation work rates than those of the uncharged TS specimen. The hydrogen-charged LS specimen has slightly higher separation work rates than those of the hydrogen-charged TS specimen.

Figures 29 and 30 show the stress triaxialities $\alpha$’s and the maximum stress triaxialities in front of the growing crack tips for the LS and TS bend specimens, respectively. Here, the black dotted lines represent the $\alpha$ distribution in front of the growing crack tips and the red solid lines represent the connections of the maximum stress triaxiality locations in front of the growing crack tips. The simulation results indicate that the maximum values of $\alpha$ increase and then decrease with increasing crack extension. The maximum values of $\alpha$ in front of the growing crack tips increase from 2.11 to 4.15, 2.10 to 3.39, 2.08 to 3.51, and 2.09 to 3.15 for the LS-2, LS-4, TS-2, and TS-4 specimens, respectively. For a reference, the stress triaxiality for the asymptotic crack tip stresses for an elastic perfectly plastic material under plane strain conditions can be calculated as 2.39 (Rice 1968).

Fig. 29

The stress triaxiality in front of the growing crack tips for the a LS-2 specimen and b LS-4 specimen

Fig. 30

The stress triaxiality in front of the growing crack tips for the a TS-2 specimen and b TS-4 specimen

The stress triaxiality can be a factor to determine the magnitude of cohesive energy as suggested by Siegmund and Brocks (2000a). In general, the cohesive energy is lower when the stress triaxiality is higher according to the Gurson material model for ductile fracture as discussed in Siegmund and Brocks (2000a). However, from the energy balance viewpoint based on the test results, the separation work rate increases and then decreases with increasing crack extension. Therefore, the stress triaxiality may not be the dominant factor to contribute to the changing separation work rate or the changing cohesive energy according to the 2-D FE analyses. The loss of plane strain conditions in the thickness direction with development of necking with increasing crack extension may be the more dominant factor for the initial increasing separation work rate or the initial increasing cohesive energy based on the results of the 3-D FE analysis of the uniform crack extension by nodal release as discussed in Sect. 5. The final decreasing separation work rates or cohesive energies may be related to the large drops of the specimen loads and increasing out-of-plane stresses near the side grooves or the increasing stress triaxialities under fully yield conditions as discussed in Sect. 5.

In summary, the near-tip maximum opening stress and separation work rate from the 2-D FE analyses by nodal release show the loading capacity of the material elements ahead of the growing crack tip and the required energy balance of the entire specimen for the stress-state dependent crack growth. The effects of the thickness, side grooves and necking give the changing near-tip maximum opening stress and separation work rate from the 2-D FE analyses by nodal release. Therefore, the effects of stress triaxiality and the thickness enter into the TSL through the changing cohesive strength and cohesive energy based on the trends of the near-tip maximum opening stress and separation work rate from the 2-D FE analyses with nodal release.