Methodology

The basis of the methodology to be applied here is as follows. It is first necessary to reflect on the tools or models that are reasonably available to use. The tools are (i) the Abaqus Finite Element Analysis (FEA) software package [69], and (ii) geometry creation and statistical analysis tools available through Python scripting [70]. Following prior representative modelling approaches (Section 2.3) reviewed above, it has been established that 2D computational specimen geometry with surface roughness and circular void porosity can be generated randomly based on generating algorithms (heuristic tools), and that these geometries can be adequately meshed and characterised. Thus, it is reasonable to expect that fair computational specimens can be created, and repeatable and statistically reliable results can be obtained from them. Any comparison of results would require assessment of the statistical variation of the “porosity volume fraction” in geometries created by the heuristic tool, and relate the geometries of the sets of results to the statistical distribution. The reader may wish to skip ahead to Section 4 to learn about the modelling methods and how this requirement can be satisfied, before returning to this present section for more detailed scrutiny.
Second, we pose some probing questions, listing all the possible effects, and then reasoning as to an appropriate approach for testing the strength and significance of the effect. This is Bacon’s “inductive reasoning” approach [1]. Using the tools and models to answer these questions through a series of computational experiments should then provide new science understanding.
The questions are:
  1. Effect of random variation – What is the nature of differences in results for models with pores created by the same heuristic tool, but with different random number seeding?
  2. Effect of length scale – What is the significance of the relative size of pores and the roughness features of the surface profile?
  3. Effect of pore size and position – What is the significance of the relative size of pores and the depth of the sub-surface region in which they appear?
  4. Effect of porosity volume fraction – What is the significance of the porosity volume fraction and how can that be characterised?
  5. Effect of porosity distribution – What is the significance of the porosity distribution and how can that be characterised?
  6. Effect of limiting scale – Should the modelling reflect the limiting scale of continuum mechanics: should a molecular dynamics approach be applied?
A number of further questions will occur readily to the critical reader, but such questions would probably address issues with the design of the computational experiments, rather than the scientific outputs themselves. A discussion of the modelling methods is given in Section 7.

Effect of random variation

The first probing question actually conceals another: there is the question of the variability in the surface profile as well as in the placing of the pores. To some extent that concealed question is already answered in that the construction of the surface shows variation along its length, so the effect of different features and the interaction between those features can be observed, (see later, for example in Figure 5). The other part of the question concerns the relative placement of the pores on the domain, and the variability of that includes the variability of juxtaposition of pores with particular surface profile features. In this way, these two questions resolve down to one computational experiment.
Resolution 1 : for each computational analysis for a given set of input data, repeat the model building using a heuristic tool, so as to produce a number of sets of results from a corresponding number of similar models. To ensure that each similar model has a correspondingly similar porosity volume fraction, the heuristic tool must generate an even distribution of porosity.

Effect of length scale

The question of length scale applies to each combination of feature size employed in the modelling. The length scale dependent features defined in the present modelling scheme are: the surface profile, the maximum pore diameter, and the position and dimensions of the defined zone area.
The surface roughness profile combines aspects of several length scales. In defining that profile, the intention was to build in a fractal-like property. Because of FEA model size limitations it is impractical to define roughness below a particular size. On the other hand, it is reasonable to consider roughness feature sizes to be similar to porosity feature sizes, so our concern is only with the relative sizes of both to within about an order of magnitude. Because the surface profile already has a fractal-like property spanning about an order of magnitude, this length scale issue is already addressed in the existing modelling approach.
Next, let us first consider the defined zone area. The defined zone height should not have a length scale effect, since it is set to be the same as the surface roughness band, and plays the same role: it enables greater variation within a model. In practice, the size of the defined zone height would play a statistical role, but this is not a question of the nature of the result to be achieved but of the precision of that result. There are two further dimensions to consider: the offset from the nominal surface, and the width. These can now be compared with the other length scale features.
The offset should be considered carefully. The main problem is that an offset is necessary to ensure that pores cannot intersect the surface. Using the heuristic model creation tool, varying the offset cannot be considered without also considering varying the maximum pore dimension, because the offset defines the allowable position of the centre of the pore, so that larger pores can extend further towards the surface than smaller ones. In view of this, perhaps the issue of relative size of the offset and the maximum pore size can be addressed by the same computational experiment, viz. : varying the maximum pore diameter.
Resolution 2a: vary the size of the maximum pore diameter by about an order of magnitude, for the same offset dimension.
Finally, let us consider the width of the defined zone area. This is similar to considering the number of pores that can be placed within the defined zone. Geometries with small pores with close packing would look similar to scaled versions of geometries with larger pores and a wider defined zone. In this case, the major difference would be in the scale of the offset and the surface roughness profile; however, those differences are separated from the left hand side of the defined zone by the presence of multiple pore features. Those features and Saint Venant’s principle would suggest a sufficient separation of detail, as to suggest that there would be little length scale interaction between defined zone width and details at or near the surface.
Resolution 2b: disregard this issue in the present study.

Effect of pore size and position

There are two parts to this issue: the first is to address the effect of large pores versus smaller ones, and the second is consider the role of pore size for pores situated deeper into the body of the material. Taking the second issue first, it should be noted that in this paper porosity is considered as a near surface phenomenon, rather than an effect on the entire material bulk [48]. As a surface phenomenon, it is reasonable to consider the presence of porosity to a maximum depth into the sub-surface, and it is also reasonable to suggest that this is achieved by making the pore diameter shrink to zero towards that limiting depth. For a thorough assessment, it might be ideal to allow the pore size to be randomly assigned, but with a distribution rule, such that the mean pore size is related to the pore depth into the sub-surface.
Resolution 3a: apply a simple linear relationship to fix pore diameter for each pore depending on pore depth. (Random assignment of pore diameter, or other interpolation schemes, could be addressed in future work.)
Having determined that the pore size at any depth in the sub-surface can be fixed in proportion to its depth, there remains only one variable, which is the maximum pore diameter.
Resolution 3b: vary the size of the maximum pore diameter by about an order of magnitude, for the same surface profile.

Effect of porosity volume fraction

In the typical manufacturing context, porosity volume fraction is a measure of the void content in a bulk of material. It is somewhat difficult to apply the same measure to porosity within the sub-surface, and to ensure that the characterisation method is meaningful: this will be discussed in Section 4.3.3. In this paper, the aim has been to ensure that the porosity is distributed as evenly as possible, using the Spacing Factor to exclude pores from approaching too closely, and by making multiple attempts to place pores.
Resolution 4: vary the value for the Spacing Factor by about an order of magnitude, for fixed maximum pore diameter.

Effect of porosity distribution

There are two ways in which the porosity distribution could be varied: systematically, or randomly. In regard to systematic variation, the present geometry creation scheme relies on using the same value of Spacing Factor for each pore within any individual model created. This means that in relative terms, the porosity volume fraction reduces with depth into the sub-surface. Varying the Spacing Factor, and/or varying the relationship between pore location and pore size, would lead to a systematic change in the local porosity distribution.
Random porosity distributions could be achieved by using smaller values for the Spacing Factor, thereby allowing closer approach, but reducing the number of trial pore placements, so that the distribution is not“fully dense” [57-58]. Such a scheme might also provide a means for characterising the nature of the porosity; however, results reported to date are qualitative rather than quantitative.
Resolution 5: postpone the consideration of the effects of systematic and random variation in porosity volume fraction for a later publication.

Continuum mechanics limit

We recognise that this is an area which deserves further consideration, particularly if it becomes clear that the mathematics of fractals becomes significant part of the developing understanding. For the moment, we consider “fractal” to be limited in length-scale to that which is feasible to model using Finite Element Analysis. We also assume that the notional “small crack” which forms the basis of fatigue analysis is related in some way to the observable length-scale features of the surface profile and sub-surface: i.e. typical distances between the bigger surface troughs, and size and spacing of pores and flaws within the sub-surface. On that basis, the continuum mechanics limit will be considered as being out of scope for this paper.