Discussion

Form of the results

The results presented here are of three different types.

Significance of modelling both surface profile and sub-surface porosity

First, there are results presented in Sub-section 4.2. In this case, three models were created from two special geometry sets: a geometry defining the surface profile, and a geometry defining one particular sub-surface porosity configuration. The results presented include both von Mises and PEEQ distributions, and show how those distributions are modified over multiple reverse loading cycles. The results tell us that if both surface profile and sub-surface porosity are taken into account in the model, then the von Mises stress and PEEQ distributions are substantively different in form, and have greater depth penetration, than for the equivalent results for models where only the surface profile or only the sub-surface porosity is modelled.

Statistical results and particular examples

The second type of result presented comes from statistical analysis of multiple geometry creations. The purpose of this computational experiment was to determine whether the heuristic tool for porosity geometry generation was creating models that had satisfactory statistical properties. Given that this is the first attempt to make a systematic approach to modelling random geometry, FEA computational costs were kept small, meaning that domain size had to be minimal, and FEA mesh refinement limited. On that basis, true “porosity”would be ill-defined, so the statistical analysis was performed on the stand-in measure of “nominal porosity” , equal to the total pore area of the model divided by the pore placement area, as illustrated in Figure 4. Those results are shown in the right hand columns of Figures 7 and 8. It is clear that for most parameters, the distributions are well-defined, and the actual geometries used in the FEA modelling are fairly well representative of the positions within the distributions. The most telling exceptions to this are as follows:
(i) The clearly skewed distribution in Figure 7(d), for maximum pore diameter equal to 100 μm, and spacing factor equal to 3. The issue here is that the maximum pore diameter is so large that very few pores can be placed. In the subsequent analysis of results, in Figure 10, it is one of these modelled geometries that gives the greatest outlier to the trend line. This particular geometry is the one with the largest“nominal porosity” , but it is only slightly greater than that for the middle geometry. Furthermore, while the difference in“nominal porosity” between the left-hand and middle geometry is much greater, the PEEQ penetration for these two models is rather similar. In any case where there is random variation, then there is a possibility for extremes to occur – in this case, because there are only a few pores (three or four), the effect of the extremes is exaggerated.
(ii) The third geometry of Figure 8(c), maximum pore diameter equal to 50 μm, and spacing factor equal to 4. This falls well to the right of the standard deviation, and of all the geometries modelled is the most like an outlier. The PEEQ penetration result for this example, is also the biggest outlier to the trend plotted in Figure 11. In this case, although the “nominal porosity” for this particular geometry is greater than those for the other two models with the same parameters, the PEEQ penetration is actually considerably lower. This indicates an effect of regularity: the only way that a higher than usual porosity can be achieved is through an arrangement of pores that comes close to a regular close packing. For that kind of regularity, there is an emergent “self-shielding” property whereby the pores themselves act as stress relieving features to their neighbours [56].

Pore placement and scaling effects

The third type of result presented is that of the multiple FEA models, showing PEEQ after five half cycles of loading. The selection of the parameters modelled, FEA meshing techniques, and the form of the results was informed in part by the earlier modelling work.
The results presented in Figures 7 and 8 provide a visual indication of the statistical placement for the 21 particular examples forming part of this study. It is assumed that for each of these geometries, the pore placement is “fully dense” , meaning that it would be impossible to find any additional site in which another pore could be sited while still obeying the pore spacing requirements. This was built into the heuristic tool, by allowing for a pragmatic number of extra tries, and can also be verified by eye by inspection of the actual models. There is a possibility that there would be non “fully dense” examples in the statistical trials.
The FEA PEEQ result figures not only indicate the placement, but also the way in which such placement leads to the particular form of the PEEQ distribution. There are clear differences in scale from one row of results to the next, but the interconnected diagonal lattice form is common to most of the figures. In some cases, the interconnectedness is more complete than in others, but where there is good connection from right to left, this seems to lead to the greater PEEQ penetration results.
For the smaller maximum pore size and smaller spacing factor results, the interconnection does not have to span the domain top to bottom in order to achieve interconnectedness from right to left. In these cases, there is more of a tendency to form a “<” shaped wedge, defined on the right hand side by some of the deeper furrows in the surface profile, and spanning chains of multiple pores, to intersect at the PEEQ penetration depth. It seems that the deepest PEEQ penetration occurs in those models for which there are near surface pores close to deeper furrows, but it should also be remembered that such a placement of those pores does influence the possible sites of neighbouring pores. In Figures 7(a) and 8(a), the particular pore configuration seems to make very little difference to the position and size of the “<” wedges or the region where the greatest PEEQ penetration occurs: these seem to be directly influenced by the positions of the deeper furrows in the surface roughness profile. The two relatively close deeper furrows in the upper third of the domain seem to promote less PEEQ penetration than the pairing that spans the lower two thirds of the domain: in both cases, the penetration depth seems to be approximately the same as the distance between the furrow pairs forming an approximate equilateral triangle. This seems to be a counter-intuitive, emergent result.
For the larger maximum pore diameter and the larger spacing factor models, there are significantly fewer pores in the models, so the formation of chains of pores cannot happen. For the larger maximum pore diameter models, Figure 7(d), the combined coincidence of there being two pores near to two of the deeper furrows seems to be the most significant factor in the PEEQ penetration depth, and possibly that is helped by the placement of a third pore near to the intersection of the PEEQ diagonals behind those two pores.
For the case of the larger spacing factor models, Figure 8(d), there are insufficient pores for interconnected chains to appear. The“nominal porosity” and the PEEQ penetration results for these three models are too similar for discussion of any further feature differences. On inspection of Figure 8(c), a somewhat more interesting picture emerges. Here the “nominal porosity” spans the distribution range well. The two left hand models show a diagonal PEEQ chain of two pores, both of which being in reasonable proximity to deeper furrows in the surface, and resulting in similar PEEQ penetration depths. In those two examples, all of the other pores are reasonably well separated from each other. The right hand example, already noted as being an outlier, shows multiple chains of pores, but none of these groupings leads to the perfect diagonal alignment that seems to be necessary for the larger PEEQ penetration achieved by the other two examples. In Sub-section 6.1.2, this effect was attributed to the higher“nominal porosity” in this example, and supposing that this was connected with a close packing effect. Looking more closely at the detail, the packing is concentrated in the upper half of the domain, with six pores arranged similarly to the spots on a die. This packing arrangement is dissimilar to the closest packing, which would be hexagonal packing; however, this square packing configuration seems to provide greater stress shielding.

Meaningful measures: PEEQ penetration and“porosity”

The difficulties in defining a measure for porosity have been discussed, and the proxy used in the data analysis, the “nominal porosity” , has been defined. The spread of results seems to increase with increasing maximum pore diameter, but it should also be remembered that for the larger pore diameters, there are fewer pores in each model, and therefore a greater variation would be expected. Because the area over which the pores are allowed to fall is poorly defined, any comparison between results from models with different maximum pore diameters and spacing factors must be considered qualitative rather than exactly quantitative.
In view of the limited number of result data points, the trend lines given in Figures 10, 11 and 12 are linear trend lines, but there is no intention to imply that there is a strictly linear relationship. Although the data is insufficient to define the relationships exactly, there is a clear trend of increasing PEEQ penetration distance with increasing maximum pore diameter, and of increasing PEEQ penetration distance with decreasing spacing factor.

Example cases and “worst cases” – assume worst case always exists!

The models presented are of limited domain size, and so the value of PEEQ penetration distance represents the value seen in the particular configuration. Had a larger domain been modelled, then there would also be a larger number of pores, and a greater opportunity for the random configuration to give rise to a somewhat larger PEEQ penetration value. The larger the modelled domain, the more likely that it will contain a configuration that leads to high PEEQ penetration depths, and the less likely that it will be entirely comprised of fortuitously aligned pores such as in the right hand model of Figure 8(c).

Mesh size effects

Since the pores are arranged randomly, it is impossible to define an entirely regular mesh. In so far as regularity is possible, the focus has been on the most critical regions: at the surface, and around the surfaces of the pores. Mesh seeding at the boundaries ensures mesh regularity for the first layer of elements adjacent to the surface. To control mesh properties deeper within the bulk of material, it is necessary to use partitioning. The mesh style around the pores was standardised, using partitioning methods described in [56]: this text also discusses the issues of mesh uniformity and adequacy of mesh choice on pore boundaries. The mesh quality between the pores and between the pores and the surface was less easy to control automatically, but each mesh was inspected, and judicious use of partitioning made to ensure that elements were of reasonable sizes and shapes.
The elements in the regions close to the boundary and between the pores are so small that it has been necessary to suppress the display of the mesh lines. As the results in the Figures 5, 6, 7 and 8 have been presented in “quilt” style, the stress or PEEQ values are shown as constant within each element. Thus, the mesh size variation of the larger elements can be seen by the jagged outline at the colour boundaries. There is some variation in the sizes of these larger elements, both within any particular model, and between models.
The error bars for penetration depth in Figures 10, 11, and 12 indicate such element size variation, and it is clear from these figures that the variation which could be ascribed to mesh choice is generally smaller than the variation between models with the same parameters, and smaller than the trend change. In view of this, it can be assumed that the mesh size control for the models presented here is adequate.

Weaknesses in the modelling that could be addressed in future

There are many aspects of the geometry creation used in this work that should be questioned. First and foremost, there are many difficulties in generating artificial random geometry while at the same time having some control over the resulting form. The methods described have attempted to provide both variation, and a controllable and quantifiable uniformity. Any improvements or changes to the geometry creation methodology would still need to demonstrate similar control.
The ideal geometry creation method might be to generate the material geometry by modelling all the essential physics of the manufacturing process. This would include fluid flow, solidification, modelling flaws and inclusions and out-gassing, and subsequent finish machining. This would be highly challenging work, and would not be a feasible approach, at least in the near future. Another approach would be to take samples of real material, and image it using X-ray Computed Tomography, and construct the CAD geometry from that data. This would be feasible but expensive. Both of these approaches would apply to a particular material for a particular manufacturing process: neither would be capable of providing a generic material model.
The algorithm process of arranging pores within the bulk of material is probably a reasonable approximation to the real emergence of pores within a material; on that basis, the pore placement algorithm could be a reasonable approach. However, close to the material surface, any real pores that form there could actually burst out of the surface and become part of the surface profile. A geometry creation algorithm that reflects real processes better might provide more realistic geometry; for example, a surface formed by pore busting, or a surface formed by the finish machining cutting through a sub-surface pore. The results from these two different model types might help determine how beneficial finish machining is for enhancing life.
The choice of circular pores is a natural and easy one: if a pore arises from trapped gas in a near constant pressure field within a near homogeneous semi-liquid material, then the pore it forms is likely to be near spherical. Whether that is the case in many typical real materials is questionable. Likewise, there are similar questions to be raised about the natural formation of an exposed surface, and how the surface profile is created based on thermal and pressure variation as the solidification takes place. The models also assume that there is no initial internal residual stress. The assumptions made here might be reasonably valid for some forms of Additive Manufacturing with post-manufacture heat treatment.
Many other researchers have focussed on the material grain structure and morphology as the driver for their modelling. The significance of the grain structure is another aspect that should be tested in future models.
The load choice and strain-hardening material model used in the Finite Element Analysis (FEA) described in this paper has been tailored to obtain significant levels of equivalent plastic strain (PEEQ) after only a few loading half cycles. To model fatigue crack initiation and propagation in a way that can be compared with test specimen data, it will be necessary to reduce load levels, analyse much higher numbers of cycles, and introduce a mechanism to represent crack propagation. These are significantly greater analysis challenges.
Finally, it has to be said that the easiest models to create and analyse are 2D models, but the real material is 3D. Creating 3D models presents some very significant challenges: much more complexity in geometry creation and control; significantly greater challenges in FEA mesh building; and, for any reasonably sized model, the computation would require High Performance Computing. For now, the results of the 2D analysis work are sufficient to be indicative and qualitative: they inform us about the effects of geometry. For more conclusive and quantitative results, 3D analysis will become necessary.

Linking these results to fatigue assessment

This paper has almost completed the full circle of the argument. We have discussed the combined role of surface roughness and sub-surface porosity, and explained how these lead to greater “penetration”of equivalent plastic strain into the sub-surface region. However, in the literature review, we discussed the suggestion by Gorelik [52], that some measure of the surface roughness could stand as a proxy for an initial crack length for fatigue life prediction using the DTDA analysis [16].
  1. We anticipated that this might provide an approach that could capture the effects of surface-breaking cracks, and other features of AM parts, so as to be able to make reliable life predictions for high duty and safety critical components.
  2. A further practical aspect of this to consider the dressing the surfaces of AM components: to provide guidelines on the appropriate depth of surface machining needed to remove imperfections, and thereby extend the component operational life.
The results that have been achieved through modelling indicate that the“PEEQ penetration” depth is a suitable and meaningful length measurement that might be taken as (or related to) EIDS when performing a durability analysis. To take this forward, it will be necessary to obtain typical surface profile and sub-surface flaw size data for the specimens used in the tests, for example those referenced in [15, 17-18], and to establish that using this value for EIDS yields conservative lives. If successful the results obtained from the present paper could be scaled to match, to provide a candidate EIDS, and hence establish whether this modelling approach has a useful predictive power.
To answer the second point, it should be remembered that surface machining is in itself a manufacturing process. On the one hand it might remove the top layer of a material, but the new surface will have a new surface profile. Furthermore, the sub-surface under that new surface might also contain defects, which might be pre-existing, or have been developed or modified by the machining process itself. Thus, the new surface and its sub-surface would need to be assessed in the same way as the original surface, and a new EIDS assessment made.

Linking these results to airworthiness requirements

The computational models developed for this paper and for the previous work on which this has been based [56, 63-64] are scale independent. In other words, the dimensions of features can all be scaled to a reasonable extent, so long as the macroscopic length scale remains representative of the everyday component size, and that the size of the smallest feature is reasonably larger than the atomic length scale limit. By working initially independently of length scale, we avoid biased thinking. Subsequent to performing the computational modelling and analysis, we have re-scaled the dimensions of the models to align with typical surface roughness size (≤ 0.1 mm) and a maximum defect size (≤ 0.1 mm) that is similar to the EIDS suggested in EZ-SB-19-01. It is these re-scaled values that are presented.
On inspection of Figures 5 and 6, and Table 3, we see the penetration depth of the equivalent plastic strain (PEEQ) into the sub-surface of the component. For this particular data, the surface roughness size is ≤ 0.1 mm, and the maximum defect size (pore diameter) is 0.05 mm. Considering surface roughness alone, the PEEQ penetration is around 0.25 mm; and considering porosity alone, it is around 0.4 mm. For the combined effect of surface roughness and porosity, the PEEQ penetration is around 0.55 mm. Furthermore the material which has undergone plastic strain is networked, meaning that these are not isolated pockets of strain, but paths in the material that represent potential failure.
Remember, that this analysis did not include element deletion or other mechanisms for modelling failure, but bulk materials will generally fail at sufficiently high levels of strain. In these models, we have also demonstrated that the strain pattern is established on the first load application, and that, while repeated load cycles increase the level of strain, there is little further change to the overall pattern of strain. In other words, for two otherwise similar materials, if one exhibits a higher strain to failure characteristic than the other then it will endure more load cycles, but will fail in a similar way. Essentially, this means that we can read across between materials and fatigue test data: it is a possible partial explanation as to why the fatigue test data collected is so consistently mapped across to the Hartman-Schijve variant of the NASGRO equation.
Now let us consider the PEEQ penetration depth. The network pattern shown in Figure 5, and also in Figures 7 and 8 for different maximum pore diameters and porosity distribution, is consistent in form and PEEQ penetration depth. If we compare the penetration depth to the EIDS of 0.01 inches (0.254 mm) suggested in EZ-SB-19-01 [6] we see a similar order of magnitude. Considering the surface roughness alone, then there is a near perfect match; however, it is not conservative to assume that there are no sub-surface defects. Even if our models are not particularly representative, they do indicate an important additional effect. Indeed, for the larger size pores (0.1 mm) and for the higher concentrations (Figure 7(d) and Table 3, 4th row), the PEEQ penetration reaches 0.73 mm. One could say that this suggests the EIDS should be increased to 0.03 inches (0.762 mm), but this would be failing to remember the scaling, and the fact that any EIDS must be determined analytically and must result in a conservative estimate of the operational life of the part. The scaling seems to be more important to the surface roughness, but there remains a difficulty in how to characterize the roughness, since it is the relative distance between neighbouring deeper furrows that seems to be important.