Adjust Material Law to Improve Necking Behavior

In order to simulate physically the contribution of each element in the necking point, it is advisable to adjust the curve by varying the Johnson-Cook coefficients in order to increase the intensity of stress at the necking point. The main result is no longer the variation of the stress/strain curve but rather the surface under the curve which characterizes the energy dissipated during the test. This energy-based approach is relevant for crash tests since the final assessment is often more significant than how it was achieved.(1)

Figure 1. Engineering Stress/Strain Curve Obtained using Adjusted Johnson-Cook Coefficients

Figure 2 compares the new yield curve with experimental data:

Figure 2. Yield Curves

Material is described in the Johnson-Cook coefficients are (Model 3):(2)

Yield stress

50 $\left[\mathrm{MPa}\right]$

Hardening parameter

350 $\left[\mathrm{MPa}\right]$

Hardening exponent

0.38

Maximum stress is set to 189 or 190 MPa (according to the parts).

The results of adjustment to the Johnson-Cook coefficients are:

Figure 3. Shell Contribution during the Necking Point Sequence (von Mises stress)

As the necking point progresses, more physical results are obtained due to the new input data of the material law coefficients having a better element contribution.

Figure 4. Variation of the von Mises Stress on Elements 110, 109, 108, 107, 11 and 106

Damage Modeling with Plastic Strain Failure

The elasto-plastic model of Johnson-Cook is used until failure, which is simulated using a plastic strain failure option. The element is deleted if the plastic strain reaches a user-defined value ${\epsilon }_{\max }$ . This damage model shows good stability. A maximum plastic strain is defined for each Johnson-Cook model:

Figure 5. ${\epsilon }_{\max }$ = 75% ; Yield Curve Close to Experimental Data:

Figure 6. ${\epsilon }_{\max }$ = 47% ; Yield Curve Adjusted with Respect to Lower Stresses:

Figure 7. ${\epsilon }_{\max }$ = 40% ; Yield Curve Adjusted with Respect to High Stresses:

Failure is reached for relatively high true strains.