investigates how finite element simulations can be used to model the tensile test curve of a PM steel, and estimate properties at different densities. A metallographic investigation of the pore and micro structures of the material is combined with a model of the strain hardening of the matrix material into a finite element model. The tensile curves, as a function of density, are then simulated using the computer model, and finally compared with experimental data.
Introduction
Along with improvements in computational power, the possibility to use simulations for material design has also improved. There are many benefits with computational materials design, including a reduction in development cost and time, and it also helps promote a better understanding of the interactions between material and structure and how to optimize materials for different performance.
Furthermore, it can also be used as a tool to answer questions about how the strength would be impacted by certain differences in the structure.
The focus of this paper is to investigate how the impact of density on the stress-strain curve in a pressed and sintered steel can be modelled using finite elements. Tensile tests of materials at different densities are combined with investigations of the pore structure into a model that is then used to simulate the material behaviour. Finally, the results from the simulations are compared to the experiments.
Material and experiments
The material used for the investigation is Distaloy AB (Fe/1.5%Cu/1.75%Ni/0.5%Mo)+0.3%C at densities of 6.3, 6.6, 6.9 and 7.2 g/cm3. Sintering was done at 1120°C for 20 min in 95/5 N2/H2+200 l/h CH4 with normal cooling. The microstructures are shown Figure 1, and the resulting properties in Table 1.
Table 1. Summary of material properties.
Series
Sintered Density [g/cm3]
Rp0.2 [MPa]
Rm [MPa]
Elongation to fracture
[%]
6.3 6.30 194.8 299.7 2.22
6.6 6.64 234.9 378.4 2.84
6.9 6.92 265.1 444.6 3.29
7.2 7.21 298.1 531.1 4.68
Figure 1. Microstructures, a. 6.3 g/cm3, b. 6.6 g/cm3, c. 6.9 g/cm3 and d. 7.2 g/cm3. Model
To set up the finite element model, it’s first necessary to have a material model for the matrix behaviour of the material. The diffusion alloyed materials typically have a very heterogeneous micro structure, as seen in Figure 1, but it was decided to model the matrix as a homogenous structure that averages the contributions from the different constituents. Furthermore, it was decided to use the Ramberg-Osgood equation for the matrix material, as a common hardening law. For the uniaxial case this equation is written as:
1 n
e p
E K
= + = +
The challenge is to translate the properties for the porous material, as given in Table 1, into the pore free matrix material behaviour. Schneider demonstrated in [1] how power law type equations can be used to model the influence of density on the properties of sintered steels. This approach uses a power law, often referred to as a Balshin or McAdam model, to capture the effect of density. Thus, the different properties can be written on the form:
( )
0 0m
Y Y
=
where index “0” refers to the property of the pore free matrix material. The pore free density was assumed to be 7.84 g/cm3. Now, it’s possible to use the experimental tensile test data to make a least squares fit of the power laws for the plastic hardening parameters in the Ramberg-Osgood equation.
Note that for the strain hardening exponent n, the strain hardening factor (n/K) was instead assumed to follow the power law since that provided a better fit. For the elastic properties the power law exponent m=3.4 was used in accordance with [2].
Table 2. Power law parameters
Young’s modulus, E E0=211.6 GPa mE= 3.40 Plastic modulus, K K0=1153.3 MPa mK= 2.83 Strain hardening factor (n/K) (n/K)0=1.37×10-4 MPa-1 mn= -3.52
By using the parameters in Table 2 it is now possible to construct a stress-strain curve for the matrix material to be used as input into the finite element model.
The next part of the model is to have a description of the porosity. In the finite element model the mesh is divided into elements representing matrix or pores, as illustrated in Figure 2. In this way the effect of density is captured in the model through the amount and sizes of pores in the material. Two different ways are used to set up the pore structure - either micrographs of a section of the material is
Figure 2. Schematic illustration of part of the FE-mesh, dark grey elements correspond to matrix material and lighter grey to pores.
In all cases the elements were triangular constant strain elements and the total mesh size 2´500´500 elements, giving an element size of 3 µm. A schematic overview of the FE-model and the boundary conditions is shown in Figure 4. At this point the model is limited to 2D, and plane strain was selected as most representative. Further details of the model are also found in [3]. For the matrix elements, a plastic von Mises material model was selected with hardening as stress strain curve according to the parameters in Table 2. For the pores, an elastic material with E=1 MPa was selected. This way “loose”
material points within a pore is automatically handled since the pores has some stiffness in the model.
However, the low value of E means that the pores do not contribute to the calculated stress-strain behaviour.
Figure 3. Measured (left) and constructed (right) pore structures.
Finally note that the simulations were run until the FE solver was no longer able to converge, which is typically the case when large local strain concentrations occurred. This means that there is so far no attempt to model the tensile strength or elongation, only the initial hardening is captured, including the yield point Rp0.2. The simulated curves stop at the point of numerical issues in the simulation. By working on the mesh and convergence a larger portion of the curve can potentially be modelled in the future, and with additional failure criteria also ultimate strength and elongation.
Figure 4. Schematic overview of the FE-model.
Results
In Figure 5 the simulated stress-strain curves for the two models are shown along with the experimental curves. Note that the simulated curves were terminated when the FE-model ran into numerical difficulties, and this does not reflect any failure behaviour of the material. Thus, only the initial hardening of the curves is compared.
Figure 5. Experimental and simulated stress-strain curves. Note that the simulated curves were cut short when the numerical model no longer converged.
As can be seen in the figure, model M (pores directly from the micrographs) follows the experimental curves quite well, except for the lowest density. Model T (constructed porosity) on the other hand consistently shows a higher hardening compared to the experiments. In Figure 6 the density dependence on the proof strength (Rp0.2) is shown. As follows from the stress strain curves, model M give a closer estimate of the experimental yield stresses as a function of density compared to model T.
In both cases the slope of the simulated curves in Figure 6 are steeper than the experimental one.
Figure 6. Density dependence on Rp0.2, models vs. experiments.
Discussion
By comparing the hardening of the two different porosity models, it is clear that they display different levels of the yield strength, where the constructed pore structure (T) gives around 20% higher values despite both models having the same amount of porosity and size distribution. The main difference between these models is the shape of the pores, with sharper corners in model M compared to the circular model T. Obviously the difference in shape will yield a difference in stress concentration as well as stress gradient around the pores, which in turn results in the difference in hardening behaviour.
Furthermore, the hardening curve can also be influenced by a softening mechanism due to micro crack formation, which is not included in the present model.
How the pores should be modelled is an open question. Two different approaches were used in this paper - a pore structure directly from micrographs of a cross section of the material (model M) and a constructed pore structure (model T). Both models have advantages as well as drawbacks. It can be argued that the micrographs give a better representation of the actual structure of the material. On the other hand, one reason to carry out simulations is to investigate materials that are not yet manufactured in reality, and when they are it will be necessary to construct the porosity.
Model M is also closer to the experimental values of the yield strength, and the stress-strain curves follow the experimental curves quite well, except for at the lowest density. However, given the plain strain conditions in the 2D finite element model, it is expected that the simulated curves should actually fall above the experimental ones due to the additional constrains limiting yielding. This is also the case for model T.
When it comes to describing the density dependence on the yielding, both models have a similar slope on the yield strength versus density curves. However, the experimental results have a slightly lower slope on the curve. The investigation spans a rather large density interval, going from 6.3 g/cm3 up to 7.2 g/cm3. This means that the characteristics of the porosity will change from mostly open for low density to closed porosity for the higher density samples, see for instance [4]. A two-dimensional model does not distinguish between these different types of porosity, and the plane strain condition mostly resembles long, straight tubes. This can be compared to for instance the difference between a spherical void compared to a long cylindrical one, which will show somewhat different stress states. It is possible that some of the density dependence on yield strength can be attributed to a change in stress state around the pores. Another argument for a change in stress state in the material as the density changes is the observation on the fracture surfaces where the amount of cleavage fracture increased somewhat as for the highest density samples compared to the lowest. This is however preliminary conclusions that need to be further evaluated.
A 2D model, as discussed in this paper, is capable of qualitatively capturing the tensile behaviour of pressed and sintered materials. However, given the limitations as discussed above, a 3D model is better suited to capture the details of the actual material behaviour. However, there are several
drawbacks with a 3D approach including the additional complexity to derive a porosity model, as well as the large increase in computational effort that is required. But with the continuous improvements in both experimental techniques and simulation power, it is expected to be realistic to move over to 3D models with time and thus increase the accuracy of the simulations.
The final point is to discuss is how the models could be extended to also include fracture. In the current setup the calculations are run until the models stop converging for numerical reasons.
However, this point does not say anything about the onset of fracture in the actual material. Since local stresses and strains are available from the simulations, and things such the shear bands forming between the pores are captured, it should be possible to extend the models to also include the onset of fracture. But fractures will initiate at a local point, as in comparison to plastic deformation that is averaged over a large material volume. Thus, accurate local fracture criteria that are valid for very small volumes are required. A fracture mechanics approach may be possible, but this requires additional investigations.
Conclusions
It has been demonstrated how simulations can be used to simulate the effect of porosity on the stress strain curves in pressed and sintered materials. The models can qualitatively capture the behaviour of the materials and effects such as the impact of density. However, there are still a number of open questions that need to be addressed, including how the porosity should be modelled, reliable fracture criteria and how to go from a 2D to a 3D model. In the end it is believed that this type of simulations will grow in importance and be used to both better understand porous materials, and as a tool for material design.
References
[1] Schneider M., Density Dependent Approximation of Stress-Strain Curves for Elastic-Plastic FEA- Calculations, proceedings of EuroPM2019 in Maastricht, 2019
[2] Sander, C., Beiss, P., Elastische Eigenschaften von Sintereisen und -stahl, Proc. Materials Week 98, Vol. 6, Symp. 8, p. 113-119; MAT INFO, Wiley VCH, Weinheim, 1999
[3] Andersson M., Finite Element Based Simulations of Mechanical Properties in Sintered Steels, proceedings of EuroPM2019 in Maastricht, 2019
[4] Danninger H. et.al., Microstructure and Mechanical Properties of Sintered Iron, Part 1: Basic Considerations and Review of Literature, Powder Metallurgy, vol. 25, no. 3, 1993
