Start of Necking Point

Necking Point Simulation

The necking point is characterized by the slope value of the true stress versus the true strain curve, which must be approximately equal to the true stress. The necking point numerically appears by continuing simulation until the condition on the slope is observed.

The results are obtained using the Johnson-Cook model 1:(1)

Figure 1. Beginning of the Necking Point using only the first coefficients of the Johnson-Cook model (a, b and n)

Figure 2. True Stress versus True Strain Curve up to the Beginning of the Necking Point

The necking point can be simulated, either by adjusting the Johnson-Cook coefficients to obtain an accurate slope, or by compelling curve with a maximum stress.

Slope Near Necking Point Simulation

By implementing an energy approach, the hardening curve can be modified to achieve an engineering curve which resembles a horizontal asymptote near the necking point with the purpose of simulating the behavior of the curve as observed in the test.

For this model, the new true stress/true strain relationship is (Johnson-Cook model 2):(2)

The results obtained with those coefficients are provided below.

Figure 3 compares the Johnson-Cook model 3 with the experiment:

Figure 3. Adjusted Engineering Stress/Strain Curve to Model the Beginning of the Necking Point

The shape of the yield curve versus the experimental data is depicted in Figure 4.

Figure 4. Yield Curves

The necking point is defined as: (3)

This condition is characterized by the intersection of the true stress versus the true strain curve with its derivate.

Figure 5. Superposition of Engineering Curve and True Curve with its Derivate

Start of Necking Point using Maximum Stress Limit

For this test, the Johnson-Cook coefficients input are those set in characterization up to the necking point, the failure effect not being taken into account (the failure plastic strain is set to zero). The beginning of the necking point is set using the choice of a maximum stress value. In comparison to the experimental results (Figure 2), the necking point is well defined for a maximum stress set at 175 $\left[\mathrm{MPa}\right]$ . The limit in stress appears on the von Mises stress versus true strain curve on elements where the necking point occurs.

The maximum true stress manages the beginning of the necking, as:

Figure 6. Engineering Stress versus Engineering Strain; Necking Point Characterization

Figure 7. Variation of the von Mises Stress with the True Strain from Shell 11

Maximum stress ${\sigma }_{\max }$ is reached for von Mises stress on shells where the necking begins. To avoid overly-high stresses after the necking point, a maximum stress factor must be set approximately equal to the true necking point stress.

The following curves show the evolution of the von Mises stress versus the true strain shell at two characteristic locations of the object (3b and 3a in Figure 8):

Figure 8. von Mises Stress Curve with a Maximum Stress Limit

The beginning of the necking point is observed following the point where the stress is equal to stress versus strain derivate .

Figure 9. Yield Curve with Maximum Stress

The yield curve is described by:(4)

The derivate of the stress is very sensitive and strongly depends on the yield curve definition. Thus, introducing the necking point into the simulation is very delicate (a small change can result in many variations). The necking point should first begin on a given element for numerical reasons.

Preferred Start of Necking Point

Experimentally, the beginning of the necking point can appear anywhere on the object. The beginning of the necking point should preferably be located on the right end elements in order to propose a methodology for this quasi-static test. If the model only uses a quarter part of the object, the necking point is found on elements 30, 125 and 78.

The beginning of the necking point is physically and numerically sensitive and can be initiated on the right elements by changing a few of the coordinates along the Y-axis of the node in the right corner (node 16) in order to decrease the cross-section and privilege the necking point in this zone. Changing the node position by 0.01 mm is enough for achieving the preferential beginning of the necking point.

Figure 10. Node 16 to be Moved

A second approach also enables the necking point to be triggered on the right end side by defining an extra part, including shells 3, 11 and 4 by using a maximum stress slightly lower than the remaining part, in order to initiate the necking point locally since the necking point stress is first reached in the elements having the lowest maximum stress value, that is shells 3, 11 and 4. This method, based on material properties, is quite appropriate for demonstrating the characterization of a material law and will thus be used in the continuation of the example.

Figure 11. Localization of the Beginning of the Necking Point According to the Models using ${\sigma }_{\max }$

The material is described as Johnson-Cook model 1:(5)

${\sigma }_{\max }$

= 174 / 175 MPa

The following curves indicate the variation of the engineering stress versus the engineering strain according to the beginning of the necking point zone and in comparison to the experiment.

Figure 12. Engineering Stress/Strain Curve for each Starting Necking Point Location

There is a fast decrease in the engineering stress after the right-end necking point. The necking point, due to the boundary conditions of the y-symmetry plane (y-translation DOF released), becomes more pronounced.

The variations in the section where the necking point is found are quite similar up to the necking point. After such point, there is a sharp surface decrease for the right-end necking point, contrary to the second case where the surface decrease is more moderate.

Figure 13. Variation of Cross Section (necking point zone)