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].
- 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.
- 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.