Study how different material models influence a cold drawing FE simulation:
- The cold drawing response of a polycarbonate material
- The force-displacement response
- The onset of necking
- The natural draw ratio
Figure 1. In this article I will show how to go from a material model, to a FE cold drawing simulation, to experimental validation.
Experimental Data for Necking during Tension
I based my study on experimental data generated by Ethan Parsons, Mary Boyce, and David Parks [Polymer, 45, (2004), 2665-2684]. They are from my old research group at MIT (and Prof. Boyce and Prof. Parks were on my thesis committee), so it is clearly a high quality paper 😎. They specifically studied the large strain tensile response of Makrolon 2608. The figures below show the neck formation and growth in one of their experiments. This type of deformation response is characteristic of all glassy polymers that undergo necking.
Figure 2. Experimental images of tension test of polycarbonate. From Parsons et al.
To perform the cold drawing FE simulation I needed an accurate material model that captures the non-linear viscoplastic response of the polycarbonate. In this case I used experimental data for Lexan 9034 from a paper by Mulliken and Boyce [J. Appl. Mechanics, 2008, Vol. 75]. I analyzed their experimental data in detail in another article (“Best Material Model for Polycarbonate at High and Low Strain Rates“).
Figure 3. The true stress-strain data was obtained by Mulliken and Boyce, the approximate engineering data was plotted by MCalibration.
Figure 4. Comparison between the experimental data and predictions from the PolyUMod TNV model.
The main reason that I selected this specific viscoplastic material model is that it contains a yield evolution feature that can accurately predict the stress softening after yielding. Most other material models cannot do that.
Necking and Natural Draw Ratio
Necking is defined as a localized reduction of the cross section in a region of a tension specimen during deformation (see Figure 2 above). Two common ways to determine if this will occur are: (1) if there is a peak in the engineering stress-strain plot; and (2) using the graphical construction of the stretch-true strain shown in the graph below.
The natural draw ratio is defined as the axial strain inside the necked region after necking has started, but before the neck has propagated all the way through the gauge section of the specimen. The figure below shows one way to graphically approximate that strain.
Figure 5. Graphical construction that can be used to approximately determine the onset of necking and the natural draw ratio from experimental data. The figure shown here was generated from the calibrated TNV model.
Cold Drawing FE Simulation 1
The calibrated material model was used to simulate the dogbone-shaped geometry that was used in Parsons’ study. I ran the FE simulation using Abaqus/Standard, although I could have used almost any other FE solver (e.g. Ansys, LS-DYNA, COMSOL Multiphysics, etc) since I used a PolyUMod material model that can be used in many different FE programs. In the simulation I used 1,260 C3D20 elements. The results are shown in the video below. You can see a nice stable neck form and grow, and the natural draw ratio corresponds to a true strain of about 0.75.
Figure 6. Video of the neck formation and growth during a tensile test on a polycarbonate specimen and the predicted force-displacement response.
It is easy to convert the FE-predicted force-displacement response to an approximate engineering stress-strain response. The engineering stress is simply the total reaction force divided by the initial cross-sectional area in the gauge section. The engineering strain is more difficult to approximate since the strain is not homogeneous in the gauge section. The figure below plots two different strain approximations: (1) the purple curve use a engineering strain defined by the applied crosshead displacement divided by the total specimen length; and (2) the black curve use an engineering strain defined by the applied crosshead displacement divided by the length of the gauge section. The true response should be somewhere between these two options. In the figure I also plotted the (approximate) engineering stress-strain curves measured by Mulliken and Boyce. The figure shows that the average response from the FE simulation is totally different than the local response from the Experiments.
Figure 7. Comparison between experimental and FE approximated engineering stress-strain curves.
Cold Drawing FE Simulation 2: Nodal Imperfections
Many researchers, including Parsons et al., introduce nodal “imperfections” in the FE mesh in order to promote the necking to start from a well-defined location. This study is exactly the same as FE Study 1, except that I displaced one line of nodes on the right hand side by a distance of 0.5% of the specimen width (see figure below).
Figure 8. Nodal displacements that were introduced to promote necking at a specific location.
As expected, in this case the neck starts at the location of the introduced geometric imperfection. Since this location is far from the shoulders of the specimen, we can clearly see that the neck initially forms as a shear band at 45° orientation. The predicted force-displacement response is identical to the results from FE Study 1, indicating that the nodal imperfections are small enough not to change the FE results. Introducing nodal imperfections is useful for forcing a neck to start at a specific location, but is not necessary to get neck formation in the FE simulation.
Figure 9. Video of the neck formation and growth during a tensile test on a polycarbonate specimen and the predicted force-displacement response.
Cold Drawing FE Simulation 3: A New Material Model
To examine the influence of the material model I modified the original PolyUMod TNV material model so that the stress drops faster after yield, and so that the stress increases more rapidly at large tensile strains. It is very easy to make these changes with the TNV model: all I needed to do was to reduce the
epsF parameter for Network 3, and increase the
C1 parameter for Network 1. See my article on the TNV model for more info about this. The predicted stress-strain response from this modified material model is shown below.
Figure 10. Comparison between Mulliken’s experimental data and predicted stress-strain data from the modified material model.
With this modified material model I expect the FE-predicted force-displacement response to drop faster after the onset of necking, and the natural draw ratio should be lower. In other words, the neck should reach the shoulders of the specimen at a lower applied displacement. The figure and video below show that this is exactly what is happening.
Figure 11. Video of the neck formation and growth during a tensile test on a polycarbonate specimen and the predicted force-displacement response.
- Finite element analysis (FEA) is a really useful tool for numerically studying cold drawing and necking of thermoplastics.
- Digital Image Correlation (DIC) is a really useful tool when performing experimental cold drawing experiments.
- The PolyUMod TNV model can capture the experimentally observed necking and cold drawing results that were measured by Parsons et al.
- It is not always necessary to introduce geometric imperfections in order to get necking in a FEA.
- The material model and specimen geometry control how much necking and cold drawing that will occur.
- Converting force-displacement data from a cold drawing experiment to a material model is more difficult and time consuming.